ILNE'S 


iACTICAlffEflMETIC 


[^         rVv        Cincinnati,  Philadelphia,  Chicago. 


• 


GIFT  OF 

|V 


THE 


PRACTICAL 

ARITHMETIC 

ON  THE 

INDUCTIVE    PLAN, 


INCLUDING 


ORAL  AND  WRITTEN  EXERCISES. 


BY 

WILLIAM  J.  MILNE,  A.  M. 

PRINCIPAL  OF  THE  STATE  NORMAL  SCHOOL,   GENESEO,   N.  Y. 


JONES  BKOTHEES  &  COMPANY: 

CINCINNATI,  PHILADELPHIA, 

CHICAGO,  MEMPHIS. 

1879, 


M-rf 


COPYRIGHT,  1877,  BY  JOHN  T.  JONES. 


ELECTROTYPED  AT 

THE  FRANKLIN   TYPE   FOUNDRY, 

CINCINNATI. 


THE  design  of  the  author  in  preparing  this  work  has  been 
to  embrace  within  moderate  compass  all  the  essentials  for 
A  PRACTICAL  COURSE  IN  ARITHMETIC,  and  to  present  every 
subject  in  such  a  manner  as  to  secure  the  highest  mental 
development  of  the  learner.  To  accomplish  these  results  the 
author  has  spent  much  time  in  investigation,  and  in  consul- 
tation with  eminent  educators  and  successful  business  men, 
and  he  believes  that  he  has  included  in  this  volume  all  the 
subjects  necessary  for  the  arithmetical  part  of  a  business 
education. 

The  method  of  introducing  each  subject  is  such  that  the 
student  is  led  to  truth  in  the  path  of  the  original  investi- 
gator—  certainly  the  most  natural  and  delightful  road  to  the 
acquisition  of  knowledge.  It  is  because  of  this  special  feat- 
ure in  connection  with  every  subject  that  the  series  has  been 
called  THE  INDUCTIVE  SERIES. 

The  work  contains  oral  and  written  exercises  sufficient  in 
number  to  enable  the  student  to  master  the  principles  un- 
derlying each  subject  and  to  give  him  facility  in  numerical 

processes. 

(iii) 

438905 


IV  PREFACE. 

In  the  problems  given  for  solution  it  has  been  the  aim  of 
the  author  to  use  the  language  of  trade,  when  no  error  is 
conveyed  thereby,  thus  accustoming  the  student  to  the  forms 
of  expression  needed  in  after  life ;  and  in  general  the  author 
has  striven  after  clearness  of  statement  rather  than  technical 
accuracy  of  expression. 

It  would  be  pedantry  to  specify  the  departments  in  which 
excellence  or  originality  may  be  found,  but  it  is  hoped  that 
a  careful  examination  will  exhibit  the  logical  sequence  of  the 
steps  in  all  the  processes,  the  perspicuity  and  accuracy  of  the 
analyses,  and  the  brevity  and  correctness  of  the  definitions, 
principles,  and  rules. 

The  author  takes  pleasure  in  acknowledging  his  indebted- 
ness to  Prof.  J.  B.  DE  MOTTE,  of  Indiana  Asbury  University, 
and  to  several  other  teachers  of  ability  and  experience,  for 
timely  and  valuable  suggestions. 

Trusting  that  the  book  will,  in  some  measure,  supply  the 
popular  demand  for  a  brief  and  comprehensive  treatise  upon 
Arithmetic,  the  author  presents  his  work  to  the  public. 

W.  J.  M. 

STATE  NORMAL  SCHOOL, 
GENESEO,  N.  Y.,  September,  1877. 


4 


GONTENTS 


f<r 
K 


NOTATION  AND  NUMERATION     7 

Arabic  Notation     .     .     .  8 

Roman  Notation    ...  17 

ADDITION 19 

SUBTRACTION 33 

MULTIPLICATION    ....  43 

DIVISION 57 

Analysis  and  Review.     .  71 

PROPERTIES  OF  NUMBERS  .  75 

Divisibility  of  Numbers.  76 

Factoring 78 

Multiplicat'n  by  Factors.  79 

Division  by  Factors    .     .  80 

Cancellation 82 

Common  Divisors  ...  84 

Multiples 88 

FRACTIONS.    ......  93 

Reduction  of  Fractions  .  96 
Addition  of  Fractions     .  104 
Subtraction  of  Fractions.  107 
Multiplication   of   Frac- 
tions       109 

Division  of  Fractions.     .  115 


PAGE 

Fractional  Forms  .     .     .  122 
Fractional    Relation    of 

Numbers 123 

Review  Exercises  .     .     .  125 

DECIMAL  FRACTIONS  .    .     .  129 

Reduction  of  Decimals  .  134 

Addition  of  Decimals     .  138 

Subtraction  of  Decimals.  139 

Multiplied  of  Decimals.  141 

Division  of  Decimals.     .  143 

Short  Processes .     .     .     .  146 

Accounts  and  Bills     .     .  151 

Review  Exercises  .     .     .  155 

DENOMINATE  NUMBERS  .    .  157 

Measures  of  Value.     .     .  158 

Reduction  Descending    .  160 

Reduction  Ascending.     .  162 

Measures  of  Space.     .     .  164 

Linear  Measures    .     .     .  165 

Surface  Measures  .     .     .  167 

Measures  of  Volume  .     .  171 

Board  Measure  ....  174 

Measures  of  Capacity.     .  175 

(v) 


VI 


CONTENTS. 


PAGE 

Measures  of  Weight   .     .178 

Measures  of  Time  .     .     .  182 

Measures  of  Angles    .     .  184 
Reduction  of  Denominate 

Fractions 186 

Review  Exercises  .     .     .  190 
Addition  of  Denominate 

Numbers 192 

Subtraction   of   Denomi- 
nate Numbers    .     .     .  194 
Multiplication     of     De- 
nominate Numbers     .  196 
Division  of  Denominate 

Numbers 197 

Longitude  and  Time  .     .  199 

Metric  System   ....  202 

PERCENTAGE 207 

Interest 218 

Method  by  Aliquot  Parts.  222 

Six  Per  Cent.  Method.     .  223 

Compound  Interest     .     .  225 

Annual  Interest.     ...  228 

Partial  Payments  ...  229 

Problems  in  Interest  .  232 


PAGE 

Commercial  Discount.     .  237 

True  Discount   ....  238 

Bank  Discount  ....  240 

Review  Exercises  .     .     .  245 

Profit  and  Loss.     .     .     .  247 

Commission 254 

Review  Exercises  .     .     .  258 

Taxes 262 

Duties 265 

Stocks 267 

Insurance 277 

Exchange 283 

Average  of  Payments.     .  290 

Average  of  Accounts  .     .  295 

PARTNERSHIP 298 

RATIO 303 

PROPORTION 306 

INVOLUTION  .    .    .    .     .    .  315 

EVOLUTION 319 

PROGRESSIONS    .    .     .    .    ,  336 

MENSURATION 341 

MISCELLANEOUS  EXAMPLES  356 

TEST  QUESTIONS     ....  363 

ANSWERS 372 


NOTATION  &  NUMERATION 


Article  1.  A  Unit  is  a  single  thing. 

2.  A  Number  is  a  unit  or  collection  of  units. 

3.  In  counting  a  large  number  of  objects  it  is  natural  to 
group  them. 

Thus,  coins  are  put  in  piles  and  the  piles  counted,  envelopes  in 
packages  and  the  packages  counted,  etc.  These  piles  and  packages 
may  themselves  be  piled  and  put  in  larger  packages,  and  the  process 
continued  indefinitely. 

4.  To  express  numbers  so  that  all  may  understand  what 
is  meant  by  the  characters  which  represent  them,  the  system 
of  grouping  by  tens  has  been  adopted.     There  are,  therefore, 
single  things,  or  units;   groups  of  ten  units,  or  tens;   groups 
containing  ten  tens,  or  hundreds,  etc. 

5.  The  method  of  grouping  by  tens  is  called  the  Deci- 
mal System. 

Decimal  is  derived  from  the  Latin  word  decem,  which  means  ten. 

(7) 


8  .        ,         NOTATION   kW>    NTJMEBATION. 


Numbers 'may 'fee  'expressed 'l>y  words  or  other  characters, 
viz:  figures  and  letters. 

6.  Notation  is  the  method  of  expressing  numbers  by 
figures  and  letters. 

The  Arabic  Notation  is  the  method  of  expressing  numbers  by  means  of 
figures.  Its  name  is  derived  from  the  Arabs,  by  whom  it  was  introduced  into 
Europe.  The  Roman  Notation  is  the  method  of  expressing  numbers  by 
means  of  letters.  It  is  so  called  because  it  was  used  by  the  ancient  Romans. 

7.  Numeration   is   the   method  of  reading  numbers 
expressed  by  figures  or  letters. 

ARABIC  SYSTEM. 

8.  In  this  system   ten  figures  are   employed   to   express 
numbers,  viz: 

Figures.    0,    1,   2,    3,    4,    5,   6,    7,     8,    9. 

Names.     Naught,  One,  Two,  Three,  Four,  Five,  Six,  Seven,  Eight,  Nine. 

Each  of  these,  except  naught,  is  called  a  significant  figure. 
Naught  is  also  called  zero  and  cipher. 

9.  By  combining  these  figures  according  to  certain  prin- 
ciples, we  can  express  any  number. 

10.  PRINCIPLE.  —  When  figures  are  written  side  by  side,  the 
one  at  the  right  expresses  units,  the  next  tens,   and  the  next 
hundreds. 

EXERCISES. 

11.  1.  In  79,  what  does  the  7  express?     What  does  the 
9  express?     Read  the  number,  beginning  at  the  left. 

2.  In  58,  what  does  the   5  express?     What  does   the  8 
express?     Read  the  number,  beginning  at  the  left. 

3.  In  740,  what  does  the  7  express?     What  does  the  4 
express?     What  does  the  0  express?     Read  the  number. 

4.  Begin  at  the  left  and  read  76,  176,  106,  360,  203. 


NOTATION   AND   NUMEBATION.  9 

12.  Figures  in  units'  place  express  units  of  the  first  order; 
those  in  tens'  place,  units  of  the  second  order;  those  in  hundreds' 
place,  units  of  the  third  order,  etc. 

13.  Numbers  between  1  ten  and  2  tens  are  named  thus: 

1  ten  and  1  unit    or  11,  eleven. 

1  ten  and  2  units  or  12,  twelve. 

1  ten  and  3  units  or  13,  thirteen. 

1  ten  and  4  units  or  14,  fourteen. 

1  ten  and  5  units  or  15,  fifteen. 

1  ten  and  6  units  or  16,  sixteen. 

1  ten  and  7  units  or  17,  seventeen. 

1  ten  and  8  units  or  18,  eighteen. 

1  ten  and  9  units  or  19,  nineteen. 

The  words  thirteen,  fourteen,  fifteen,  etc.,  mean  three  and  ten,/ow 
and  ten,  five  and  ten,  etc. 

14.  The  units  of  the  second  order  are  named  as  follows : 


2  tens  or  20,  twenty. 

3  tens  or  30,  thirty. 

4  tens  or  40,  forty. 

5  tens  or  50,  fifty. 


6  tens  or  60,  sixty. 

7  tens  or  70,  seventy. 

8  tens  or  80,  eighty. 

9  tens  or  90,  ninety. 


The  suffix  ty  means  ten.     Thus,  forty  means  four -tens,  etc. 

The  other  numbers  between  20  and  100  are  read  without 
the  word  and  between  the  tens  and  units.  Thus,  27  is  read 
twenty-seven,  instead  of  twenty  and  seven. 

EXERCISES. 

15.  Read  the  following: 

28     99     43  73  67  41 

64     83     75  86  45  31 

39     78     60  51  32  92 

47     32     21  55  82  25 


10  NOTATION   AND   NUMERATION. 

16.  Express  in  figures  the  following: 


Three  tens  and  eight  units. 
Four  tens  and  seven  units. 
Two  tens  and  two  units. 
One  ten  and  three  units. 
Six  tens  and  nine  units. 


Five  tens  and  seven  units. 
Eight  tens  and  one  unit. 
Seven  tens  and  three  units. 
One  ten  and  eight  units. 
Four  tens  and  two  units. 


Three  units  of  the  second  order,  six  of  the  first  order. 
Two  units  of  the  second  order,  four  of  the  first  order. 
Write  all  the  numbers  between  10  and  20.     Between  30 
and  50.     Between  70  and  90. 

17.  In  reading  a  number  expressed  by  three  figures,  the 
tens   are   read  after   the  hundreds  without    the  word  and. 
Thus,  235  is  read  two  hundred  thirty-five  instead  of  two 
hundred  and  thirty-five. 

18.  Read  the  following: 

746  932  786  849  534 

678  453  777  391  585 

963  378  243  873  412 

531  217  918  855  248 

19.  Express  in  figures  the  following: 

Two  hundreds,  three  tens,  five  units. 

Six  hundreds,  two  tens,  nine  units. 

Four  hundreds,  one  ten,  eight  units. 

Three  units  of  the  third  order,  six  of  the  second  order. 


Three  hundred  eighteen. 
Eight  hundred  thirty. 
Four  hundred  four. 
Six  hundred  eighty-one. 
Seven  hundred  seventy. 
Seven  hundred. 


Seventy. 

Seven. 

Seven  hundred  six. 

Six  hundred  forty. 

Two  hundred  six. 

One  hundred  eleven. 


Seven  hundred  seventy-seven. 


NOTATION    AND   NUMERATION.  11 

From  the  previous  examples  we  deduce  the  following  gen- 
eral principles: 

20.  PRINCIPLES. — 1.    The  representative  value  of  a  figure  is 
increased  tenfold  by  each  removal  one  place  to  the  left,  and  de- 
creased tenfold  by  each  removal  one  place  to  the  right. 

2.   The  figure  0  is  used  to  give  significant  figures  their  positions. 

21.  In  reading  numbers  a  new  name  is  given  the  order  of 
units  next  higher  than  hundreds  of  any  denomination.     Thus, 
the  order  next  higher  than   hundreds   is  called   thousands, 
that  next  higher  than  hundreds  of  thousands,  millions,  etc. 
Therefore   each  denomination   can  have  but  three  orders  of 
units. 

22.  A  Period  is  a  group  of  figures  containing  the  hun- 
dreds, tens,  and  units  of  any  denomination. 

The  present  system  of  notation  is  illustrated  by  the  following 

TABLE. 
PERIODS.          6th.       5th.        4th.        3d.         2d.        1st. 

CQ 

NAMES          I*        |          3          £         J 
PERIODS.  3^3  .•§ 

3  ~3  fi  fi 


02  in.  in.  02  co  cc 

Q}  <3^  Q}  Q}  QP  Q^ 

ORDERS.  ^  3       ^  ^       m  «       w  «       ^  * 


30,230,160,700,401,690. 

This  number  is  read  thirty  quadrillion,  two  hundred  thirty 
trillion,  one  hundred  sixty  billion,  seven  hundred  million,  Jour 
hundred  one  thousand,  six  hundred  ninety. 


12 


NOTATION   AND    NUMERATION. 


1.  In  reading  numbers,  the  name  of  units' period  is  omitted. 

2.  Each  period,  except  the  highest,  must  contain  three  figures. 

3.  The  periods  are  separated  from  each  other  by  commas. 

23.  The  periods   above  Quadrillions,  in   their   order,   are 
Quintittions,  Sextillions,  Septillions,  etc. 

24.  Give  the  number  of  each  of  the  following  periods : 


Millions. 
Thousands. 


Trillions. 
Units. 


25.  Give  the  names  of  the  following : 


5th  period. 
3d   period. 


2d   period. 
4th  period. 


Billions. 
Quadrillions. 

1st  period. 
6th  period. 


26.  Repeat  in  order  the  names  of  the  periods  from  : 


Units  to  billions. 
Units  to  quadrillions. 
Thousands  to  trillions. 


Billions  to  units. 
Quadrillions  to  units. 
Millions  to  thousands. 


27.  Copy  and  point  off  into  periods: 


1.  46825. 

2.  239746. 

3.  180040. 

4.  14168843. 


5.  38420058. 

6.  33468204. 

7.  8438206. 

8.  436784. 


9.   5284325684. 

10.  7932468512. 

11.  83749275867. 

12.  1423789276586. 


13.  How  many  thousands  are  there  in  the  first  number? 

14.  How  many  thousands  in  the  second  number? 

15.  How  many  billions  in  the  next  to  the  last  number? 

16.  How  many  trillions  in  the  last  number?     How  many 
billions?    How  many  millions?    How  many  thousands?    How 
many  units? 

17.  Point  off  into  periods,  and  name  in  order,  the  billions, 
millions,  thousands,  and  units  of  the  next  to  the  last  number. 

18.  Point  off  into  periods,  and  name  in  their  order,  the 
periods  composing  the  12th  number. 

19.  In  like  manner  point  off  and  read  each  of  the  numbers. 


NOTATION   AND    NUMERATION.  13 

28.  Write  in  figures: 

1.  Thirty-four  billion,  eighteen  thousand,  forty. 

PROCESS.  ANALYSIS. — Since  the  highest  period  is  bill- 

ions, which  occupy  the  fourth  period,  we  make 
?      co     Si      TH       ^our  sPaces  f°r  the  periods.    We  write  34  in  the 
fourth   period,   thus  expressing   the  billions  of 


34 


000 


018 


040 


the  given  number;  18  in  the  second  period,  thus 


expressing  the  thousands;    and  40  in   the  first 
Or,  period,  thus  expressing  the  units.     Since  every 

34  000  018  040     Peri°d  except  the  highest  must  contain   three 

figures,  we  fill  the  vacant  places  with  ciphers. 
As  soon  as  possible  use  commas  instead  of  the  lines,  and  cease  to 
write  both  the  number  and  name  of  the  periods. 

Write  in  figures,  and  read  the  number: 

2.  Thirty-six  in  the  3d  period,  two  hundred  eighteen  in 
the  2d,  eight  hundred  forty-six  in  the  1st. 

3.  Eighty-four  in  the  4th  period,  five  hundred  forty  in 
the  3d,  six  hundred  in  the  2d,  forty  in  the  1st. 

4.  Two  hundred  one  in  the  5th  period,  seventy-five  in  the 
4th,  five  hundred  sixty-two  in  the  3d,  twelve  in  the  2d,  one 
in  the  1st. 

5.  Sixty  in  the  5th  period,  four  hundred  two  in  the  4th, 
three  hundred  thirty-three  in  the  3d,  two  hundred  in  the 
2d,  one  hundred  eleven  in  the  1st. 

Write  in  figures: 

6.  Seventy-three  million,  two  hundred  fourteen  thousand, 
seventy. 

7.  Eighty  billion,  forty  million,  six  hundred  twelve  thou- 
sand, seven  hundred  eighty-eight. 

8.  Two  hundred   twenty-five   million,    six  hundred  forty- 
one  thousand,  three  hundred  fifty-one. 

9.  Three  hundred  fifty-four  billion,  six  hundred  four  mill- 
ion, eight  hundred  ninety-two  thousand,  thirty-six. 


14 


NOTATION   AND   NUMERATION. 


29.  RULE  FOR  NOTATION. — Begin  at  the  left  and  write  the 
hundreds,  tens,  and  units  of  each  period  in  their  proper  order, 
putting  ciphers  in  all  vacant  places  and  periods. 

While  writing,  separate  each  period  from  the  next  by  a  comma. 

30.  RULE  FOR  NUMERATION. — Begin  at  the  right  and  sep- 
arate the  numbers  into  periods  of  three  figures  each. 

Begin  at  the  left  hand  and  read  each  period  as  if  it  stood 
alone,  adding  its  name. 


EXERCISES. 


31.  Copy,  point  off,  and  read: 


1.  116234 

8.  141120. 

15.     7640. 

2.  65231. 

9.  101207. 

16.    800900. 

3.  20703. 

10.  68978. 

17.   2568242. 

4.  71005. 

11.  72020. 

18.   1008003. 

5.   3104. 

12.  80001. 

19.  212375647. 

6.  48000. 

13.  857000. 

20.  609003588. 

7.  60029. 

14.  91029. 

21.  897856846. 

32.  Write  in  figures,  and  read: 

22.  Two  hundred  in  the  1st  period. 

23.  Sixty  in  the  2d  period,  two  in  the  1st. 

24.  Seven  hundred  in  the  3d  period. 

25.  Two  hundred  thirty  in  the  3d  period,  sixty  in  the  1st. 

26.  Eighty-one  in  the  4th  period,  five  hundred  one  in  the 
3d,  seven  in  the  2d,  twelve  in  the  1st. 

27.  Thirty  in  the  5th  period,  six  hundred  three  in  the  1st. 

28.  Seven  hundred  in  the  5th  period,  eighty  in  the  4th. 

29.  Eight  in  the  4th  period,  seven  in  the  3d,  fourteen  in 
the  2d,  and  ten  in  the  1st. 

30.  Fifteen  in  the  6th  period,  eighteen  in  the  4th,  two 
hundred  seven  in  the  3d,  and  eighty-one  in  the  1st. 


NOTATION   AND   NUMERATION.  15 

33.  Copy,  point  off,  and  read: 


1.  60701892. 

2.  50607801. 

3.  600000. 

4.  49000000. 

5.  593006070500. 

6.  19019000190019019. 


7.  163194568. 

8.  3050050183. 

9.  5000204. 

10.  594900. 

11.  12000012. 

12.  200798013400019. 


13.  2125'06067093012063067. 

34.  Write  in  figures: 

14.  Two  in  the  3d  period,  sixty  in  the  2d,  one  hundred 
fifty-three  in  the  1st. 

15.  Sixty  in  each  of  the  4th,  3d,  2d,  and  1st  periods. 

16.  60  million,  200  thousand,  500. 

17.  402  billion,  348  million,  213  thousand,  20. 

18.  78  trillion,  640  billion,  9  million,  6  thousand,  16. 

19.  6  billion,  542  million,  25. 

20.  Six  billion,  five  hundred  forty-two  million,  twenty-five. 

21.  Four  hundred  two  billion,  three  hundred  forty-eight 
million,  two  hundred  thirteen  thousand,  twenty. 

22.  Five  million,,  two  huiidred  sixty-eight  thousand,  nine 
hundred  forty-nine. 

23.  Two  hundred  million,  three  hundred  thousand,  eight 
hundred. 

24.  Twenty-nine  billion,  five  hundred  ninety-nine  million, 
six  hundred  one. 

25.  Four   trillion,    five    hundred    fifty-eight    million,    two 
hundred  forty-four  thousand,  seventy. 

26.  Thirty-two   billion,   sixty-one    million,    three    hundred 
forty-three  thousand,  four  hundred  four. 

27.  Five  hundred  fifty-five  million,  seven  hundred  seventy- 
seven  thousand,  six  hundred  sixty-nine. 

28.  Eight  hundred  six  billion,  seventy  million,  three  hun- 
dred eighty-five  thousand,  two  hundred  six. 


16  NOTATION   AND   NUMERATION. 

29.  Nine  hundred  forty-one  trillion,  one  hundred  sixteen 
thousand,  twenty-two. 

30.  Twenty-three    billion,    twenty-three    million,    twenty- 
three  thousand,  twenty-three. 

31.  Six  hundred  thousand,  seventy-five. 

32.  Twelve  billion,  eight    million,  nine    hundred    eighty- 
eight  thousand,  thirteen. 

33.  Twenty-nine  quadrillion,  seven  hundred  fifty-seven  trill- 
ion,   four   hundred    eighty   million,   thirteen    thousand,   five 
hundred  sixty-five. 

NOTATION  AND  NUMERATION  OF  U.   S.   MONEY. 

35.  The  currency  of  the  United   States  has  a  Decimal 
System  of  notation,  thus: 

10  mills  make  1  cent. 
10  cents  make  1  dime. 
10  dimes  make  1  dollar. 

36.  The  Sign  of  Dollars  is  $.     It  is  written  before 
the  number. 

Thus,  $16  is  read,  sixteen  dollars. 

37.  In  writing  decimal  currency  a  mark  called  the  deci- 
mal point  is  placed  before  cents  and  mills. 

38.  Cents  occupy  the  first  two  places  at  the  right  of  the 
decimal  point,  and  mills  the  third. 

Thus,   $7.584  is  read,  seven  dollars,  fifty -eight  cents,  four  mills; 
is  read,  sixty-nine  cents,  four  mills. 


39.  If  the  number  of  cents  is  less  than  ten,  write  a  cipher 
in  the  first  place  at  the  right  of  the  decimal  point. 

Thus,  five  dollars,  eight  cents,  is  written,  $5.08;  three  dollars,  seven 
cents,  $3.07. 


NOTATION   AND   NUMERATION. 


17 


40.  Read  the  following: 


$6.85 

$7.843 

$12.056 

$31.095 

$24.055 

$20.20 

$28.075 

$40.04 

$606.952 

$500.50 

$2103.094 

$7000.16 

$20000. 

$6001.101 

$300.416 

$212012.12 

$695.955 

$200.204 

$613.495 

$211.12 

$69.69 

$203.033 

$216.16 

$75.25 

41.  Write  the  following: 

1.  Two  dollars,  twenty-three  cents,  five  mills. 

2.  Two  hundred  two  dollars,  two  cents,  five  mills. 

3.  One  hundred  twelve  dollars,  twenty-five  cents. 

4.  Six  hundred  two  dollars,  nine  cents. 

5.  Twenty  thousand  dollars,  thirty-two  cents. 

6.  Twelve  million,  seven  hundred  thousand  dollars. 

7.  Six  million  dollars,  eighty-eight  cents. 

8.  Twelve  thousand  three  hundred  dollars,  fifteen  cents. 


ROMAN   SYSTEM. 

42.  In  this  system  seven  letters  are  used  to  express  num- 
bers, viz: 

Letters.     I,    V,    X,     L,       C,       D,        M. 

Values.      1,    5,     10,     50,     100,     500,    1000. 

By  combining  these  letters  according  to  certain  principles 
any  number  can  be  expressed. 

PRINCIPLES. — 1.    When  a  letter  is  repeated  its  value  is  re- 
peated. 

Thus,   I  represents  1;  II,  two;  III,  three;  X,  ten;  XX,  twenty; 
XXX,  thirty  ;  C,  one  hundred ;  CCC,  three  hundred. 


18 


NOTATION   AND    NUMERATION. 


2.  When  a  letter  is  placed  before  another  of  greater  value  its 
value  is  to  be  taken  from  that  of  the  greater. 

Thus,  I  represents  one  and  V  five,  but  IV  represents  four;   IX, 
nine;  XIX,  nineteen;  XL,  forty;  XC,  ninety. 

3.  When  a  letter  is  placed  after  another  of  greater  value  their 
values  are  to  be  united. 

Thus,  XV  represents  fifteen;  LX,  sixty;  LXXX,  eighty;  DC,  six 
hundred ;  MD,  fifteen  hundred. 

4.  A  bar  placed  over  a  number  increases  its  value  a  thousand- 
fold. 

Thus,  V  represents  five ;   V,  five  thousand ;   LX,  sixty ;   LX,  sixty 
thousand ;    M,  one  thousand ;    M,  one  million. 

TABLE. 


I   

.  .  .  .    1 

XIV  .  .  . 

...  14 

LX  

60 

II  

.  .  .  .     2 

XV 

.  .  15 

LXX 

70 

Ill  .  .    . 

.     .  .    3 

XVI 

.  16 

LXXX 

80 

IV    .... 

.  .  .  .    4 

XVII.  .  . 

.  .  .  17 

XC  

90 

V     .... 

.  .  .  .    5 

XVIII 

18 

c 

100 

VI    .... 

.  .  .  .    6 

XIX  .  .  . 

.  .  .  19 

cc  

200 

VII  .... 

.  .  .  .    7 

XX     .  .  . 

...  20 

CCL  

250 

VIII   .  .  . 

.  .  .  .    8 

XXI  .  .  . 

...  21 

CCCC 

400 

IX    .... 

.  .  .  .     9 

XXIX 

...  29 

D 

500 

X     .... 

.  ...  10 

XXX    .  . 

...  30 

DCC 

700 

XI    .... 

.  ...  11 

XXXIV  . 

...  34 

M 

1000 

XII  .... 

.  ...  12 

XL  .... 

...  40 

MMM 

3000 

XIII   . 

.  13 

L  . 

.  50 

MDCCCLXXX 

1880 

Read  the  following  numbers: 

XV;  XXIV;  XXXIX;  XL;  XLIX;  JXCIX;  LXXVII; 
CCCLXXXIX;  DCCXXXVI;  VDLV;  DLDC;  CCXDVI; 
LXXMMMDCCCXCIX;  MDXCVDCCCLXIV. 

Express  the  following  numbers  by  Roman  Notation: 
15,  18,  27,  81,  95,  86,  534,  684,  1050,  8004,  7000,  75869, 


IND  UCTIVE     EXERCISES. 

43.    1.  How  many  are  2  pears  and  1  pear?     2  pears  and 
2  pears? 

2.  How  many  are  3  leaves  and  2  leaves?     3  leaves  and  3 
leaves?     How  many  are  3  and  1?     3  and  2?     3  and  3? 

3.  Jane  has  3  apples  and  Mary  has  4  apples.     How  many 
apples  have  both?     How  many  are  3  and  4?     4  and  3? 

4.  George  gave  me  2  apples  and  Mary  gave  me  4.     How 
many  apples  did  both  give  me?     How  many  are  4  and  2? 
2  and  4? 

5.  A  farmer  had  2  horses  and  bought  6  more.     How 
many  horses  had  he  then?     How  many  are   2  and   6?     6 
and   2? 

6.  Henry  paid  5  cents  for  a  pencil  and  7  cents  for  a 
writing-book.     How  many  cents  did  he  pay  for  both?     How 
many  are  5  and  7?     7  and  5? 

7.  If  a  barrel  of  flour  is  worth  $6,  and  a  cord  of  wood 
$4,  how  much  are  both  worth?     How  many  are  6  and  4? 

8.  A  man  plowed  8  acres  of  land  in  one  week  and  6  acres 
the  next  week.     How  many  acres  did  he  plow  in  both  weeks? 

9.  On  the  Fourth  of  July,  Ned  spent  10  cents  for  fire- 
crackers and  6  cents  for  torpedoes.     How  many  cents  did  he 
spend  for  both? 

10.  Harry  is  6  years  old  and  his  sister  is  four  years  older. 
How  old  is  his  sister?     How  many  are  6  and  4? 

(19) 


20  ADDITION. 

11.  At  Christmas,  Horace  received  9  gifts  from  his  par- 
ents,  and  4  from  other  friends.     How  many  gifts  did  he 
receive? 

12.  A  certain  house  has  5  windows  in  one  side  and  7  in 
another.     How  many  windows  in  the  two  sides? 

13.  How  many  are  5  oranges  and  4  oranges?     6  boys 
and  3  boys?     5  horses  and  6  cents? 

14.  Why  can  you  not  tell  how  many  5  horses  and  6  cents 
are? 

15.  Why  can  you  tell  how  many  5  oranges  and  4  or- 
anges are? 

Numbers  that  express  things  of  the  same  name  are  called 
Like  Numbers. 

16.  What  kind  of  numbers  only  can  be  united? 

DEFINITIONS. 

44.  Addition    is    the    process    of  finding    a    number 
which  shall  be  equal  to  two  or  more  given  numbers. 

45.  The  Sum  or  Amount  is  the  result  obtained  by 
adding. 

46.  The  Sign  of  Addition  is  an  upright  cross:   +. 
It  is   called  plus,   and   is   placed   between   numbers   to   be 
added. 

Thus,  3  +  4  is  read  3  plus  4,  and  means  that  3  and  4  are  to  be 
added. 

47.  The  Sign  of  Equality  is  two  short  horizontal 
lines:   =.     It  is  read  equals,  or  is  equal  to. 

Thus,  3  +  4  =  7,  is  read  3  plus  4  equals  7. 

The   expression   3  +  4  =  7,   or   any  other  expression   of 
equality,  is  called  an  Equation. 


ADDITION. 


21 


48.  PRINCIPLES. — 1.   Only  like  numbers  can  be  added. 
2.   The  sum  and  numbers  added  must  be  like  numbers. 

TABLE. 


1  +  1=  2 

1  +  2=  3 

1  +  3=  4 

1  +  4=  5 

1+   5=  6 

2  +  1=  3 

2  +  2=  4 

2  +  3=  5 

2  +  4=  6 

2+   5=  7 

3  +  1=  4 

3  +  2=  5 

3  +  3=  6 

3  +  4=  7 

3+   5=  8 

4  +  1=  5 

4  +  2=  6 

4  +  3=  7 

4  +  4=  8 

4+   5=  9 

5  +  1=  6 

5  +  2=  7 

5  +  3=  8 

5  +  4=  9 

5+  5  =  10 

6  +  1=  7 

6  +  2=  8 

6  +  3=  9 

6  +  4  =  10 

6+   5  =  11 

7  +  1=  8 

7  +  2=  9 

7  +  3  =  10 

7  +  4=11 

7+   5  =  12 

8  +  1=  9 

8  +  2  =  10 

8  +  3  =  11 

8  +  4=12 

8+'  5  =  13 

9  +  1  =  10 

9  +  2  =  11 

9  +  3  =  12 

9  +  4=13 

9+   5  =  14 

1  +  6=  7 

1  +  7=  8 

1  +  8=  9 

1  +  9  =  10 

1  +  10  =  11 

2  +  6=  8 

2  +  7=  9 

2  +  8  =  10 

2  +  9  =  11 

2  +  10  =  12 

3  +  6=  9 

3  +  7  =  10 

3  +  8  =  11 

3  +  9  =  12 

3  +  10=13 

4  +  6  =  10 

4  +  7  =  11 

4  +  8  =  12 

4  +  9  =  13 

4  +  10=14 

5  +  6  =  11 

5  +  7  =  12 

5  +  8  =  13 

5  +  9  =  14 

5  +  10=15 

6  +  6  =  12 

6  +  7  =  13 

6  +  8  =  14 

6  +  9  =  15 

6+10  =  16 

7  +  6  =  13 

7  +  7  =  14 

7  +  8  =  15 

7  +  9  =  16 

7  +  10=17 

8  +  6  =  14 

8  +  7  =  15 

8  +  8  =  16 

8  +  9  =  17 

8  +  10=18 

9  +  6  =  15 

9  +  7  =  16 

9  +  8  =  17 

9  +  9=18 

9  +  10  =  19 

CASE    I. 
49*  To  add  single  columns. 

1.  A  grocer  sold  8  pounds  of  sugar  to  one  man  and  7  pounds 
to  another.     How  many  pounds  did  he  sell  to  both? 

ANALYSIS. — Since  he  sold  8  pounds  to  one  man  and  7  pounds  to 
another,  to  both  he  sold  the  sum  of  8  pounds  and  7  pounds,  which 
is  15  pounds. 

2.  A  man  rode  7  miles  the  first  hour  and  6  miles  the  sec- 
ond hour.     How  far  did  he  ride  in  the  two  hours? 


22  ADDITION. 

3.  On  one  tree  are  8  birds,  and  on  another  4  birds.     How 
many  birds  are  there  on  both? 

4.  Carl  earned  $2  in  May,  $5  in  June,  and  $4  in  July. 
How  much  did  he  earn  in  the  three  months? 

5.  I  gave  6  nuts  to  one  boy,  5  to  another,  and  3  to  an- 
other.    How  many  nuts  did  I  give  to  all? 

6.  I  paid  5  cents  for  paper,  3  cents  for  pens,  and  5  cents 
for  ink.     How  much  did  I  pay  for  all? 

7.  A  lemon  cost  5  cents,  an  orange  6  cents,  and  a  pine- 
apple 8  cents.     What  did  they  all  cost? 

8.  Esther  gave  her  teacher  5  pinks,  7  roses,  and  4  pan- 
sies.     How  many  flowers  did  she  give  her? 

9.  James  shot  9  birds,   Henry  shot  6,   and  William  5. 
How  many  did  they  all  shoot? 

10.  A  woman  picked  9  quarts  of  blackberries  one  morn- 
ing, while  her  son  picked  3  quarts.     How  many  quarts  did 
both  pick? 

11.  James  solved   6  examples,   John  5,  William   8,   and 
Henry  7.     How  many  examples  did  they  solve? 

12.  One  boy  picked  6  quarts  of  cherries,  another  4  quarts, 
another  5  quarts.     How  many  quarts  did  they  all  pick? 

13.  I  gathered  from  one  pear-tree,  this  year,  2  bushels  of 
fruit,  from  another  4  bushels,  from  another  3  bushels,  and 
from  another  2  bushels.     How  many  bushels  did  I  gather 
from  these  four  trees? 

14.  A  merchant  sold  from  a  piece  of  cloth,  3  yards  at  one 
time,  6  yards  at  another,  8  yards  at  another,  and  5  yards  at 
another.     How  many  yards  did  he  sell  in  all? 

15.  A  man  picked  8  barrels  of  apples  on  Monday,  6  bar- 
rels on  Tuesday,  4  barrels  on  Wednesday,  and  5  barrels  on 
Thursday.     How  many  did  he  pick  altogether? 

16.  Henry  learned  7  verses   of  poetry  on  one  day,  5  on 
another,  6  on  another,  and  8  on  another.     How  many  verses 
did  he  learn  in  the  four  days? 


ADDITION.  23 

17.  A  man  paid  $9  for  a  coat,  $4  for  pants,  and  $2  for  a 
hat.     How  much  did  he  pay  for  all? 

18.  In  a  garden  there  are  8  apple-trees,  7  plum-trees,  and 
9  peach-trees.     How  many  trees  are  there  in  the  garden? 

19.  There  are  4  boys  and  7  girls  in  one  class,  and  6  boys 
and  8  girls  in  another.     How  many  pupils  in  both  classes? 

20.  Homer  paid  8  dollars  for  a  fur  cap,  and  5  dollars  for 
a  pair  of  skates.     How  much  did  both  cost  him  ? 

21.  A  boy  gathered  nuts  for  three  days.     The  first  day  he 
brought  home  8  quarts,  the  next  day  7  quarts,  the  next  day 
9  quarts.     How  many  quarts  did  he  bring  home  ? 

22.  Repeat  the  addition  table  of  ones.      Of  twos.      Of 
threes.     Of  fours.     Of  fives.     Of  sixes.     Of  sevens.     Of 
eights.     Of  nines.     Of  tens. 

23.  Count  by  2's  from  0  to  20;  thus:  0,  2,  4,  6,  8,  10, 
12,  etc. 

24.  Count  by  3's  from  2  to  26.     From  26  to  41. 

25.  Count  by  4's  from  0  to  36.     From     5  to  53. 

26.  Count  by  5's  from  3  to  43.     From     7  to  72. 

27.  Count  by  6's  from  0  to  42.     From     4  to  46. 

28.  Count  by  7's  from  4  to  39.     From  11  to  60. 

29.  Count  by  8's  from  2  to  58.     From     7  to  63. 

30.  Count  by  9's  from  7  to  70.     From     8  to  71, 


WRITTEN    EXERCISES. 

50.    1.  What  is  the  sum  of  5,  4,  7,  and  6? 

PROCESS.  ANALYSIS. — We  write  the  numbers  to  be  added  in  a 

5  column,  and  begin  at  the  bottom  to  add;  thus:  6,  13,  17, 

22;  and  write  the  sum  beneath.     To  prove  the  work  we 
-  may  begin  at  the  top  and  add  downwards.     If  the  result 

agrees  with  the  one  formerly  obtained  the  work  is  proba- 
bly correct.     In  adding  say,  6,  13,  17,  etc.,  instead  of  6 
22    Sum.      and  7  are  13.  and  4  are  17,  etc. 


24  ADDITION. 

Copy,  add,  and  prove: 


(2-) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

5 

6 

5 

6 

7 

8 

3 

.  7 

6 

4 

3 

9 

4 

8 

2 

3 

8 

8 

2 

1 

3 

4 

5 

7 

(8.) 

(9.) 

(10.)       (11.) 

(12.) 

(13.) 

8 

6 

5 

8 

9 

2 

7 

4 

4 

7 

7 

5 

6 

3 

3 

9 

8 

7 

3 

0 

2 

8 

9 

9 

5 

3 

1 

8 

8 

3 

51.  Kequired  the  sum  of  the  following: 


14.  6,  7,  5,  3,  2,  4,  and  5. 

15.  8,  2,  0,  3,  3,  2,  and  4. 

16.  5,  6,  7,  6,  4,  2,  and  8. 

17.  3,  2,  6,  5,  8,  7,  and  9. 

18.  8,  3,  0,  5,  3,  8,  and  2. 

19.  7,  6,  6,  4,  3,  6,  and  3. 


20.  7,  8,  8,  9,  0,  3,  and  3. 

21.  8,  9,  7,  8,  5,  8,  and  2. 

22.  7,  6,  5,  4,  3,  2,  and  1. 

23.  5,  4,  4,  3,  2,  6,  and  7. 

24.  4,  3,  4,  5,  6,  8,  and  8. 

25.  3,  6,  8,  6,  7,  0,  and  5. 


26.  There  are  8  chickens  in  one  coop,  9  in  another,  7  in 
another,  and  5  in  another.     How  many  chickens  are  there 
in  all  the  coops? 

27.  My  father  has  5  horses,  9  cows,  7  sheep,  and  3  pigs. 
How  many  animals  has  he  in  all? 

'28.  A  man  walked  from  A  to  B  in  four  hours.  He  went 
4  miles  the  first  hour,  3  miles  the  second  hour,  5  miles  the 
third  hour,  and  6  miles  the  fourth  hour.  What  was  the 
distance  between  the  two  places? 

29.  A  house  had  8  windows  on  the  east  side,  7  on  the 
west,  and  9  on  the  south.     How  many  were  there  in  all? 


ADDITION.  25 

CASE    II. 
52.  To  add  several  columns. 

1.  Count  by  10's  from   7  to   107;  thus,  7,  17,  27,  37, 
47,  etc. 

2.  Count  by  10's  from  5  to  95.     From  9  to  79. 

3.  Count  by  20's  from  5  to  85.     From  9  to  89. 

4.  Add  2  to  each  of  the  series  of  numbers  6,  16,  26,  etc., 
to  76. 

5.  Add  3  to  each  of  the  series  of  numbers  from  8,  18, 
etc.,  to  88. 

6.  A  gentleman  paid  $7  for  a  hat,  $8  for  a  vest,  and  $13 
for  pantaloons.     How  much  did  he  pay  for  all? 

ANALYSIS. — Since  he  paid  $7  for  a  hat,  $8  for  a  vest,  and  $13  for 
pantaloons,  for  all  he  paid  the  sum  of  $7,  $8,  and  $13,  or  $28. 

7.  James  gave  25  cents  to  his  brother  and  20  cents  to  his 
sister.     How  much  did  he  give  to  both?     25  and  20  are  how 
many? 

8.  Horace  earned  35  cents  on  Monday,  20  cents  on  Tues- 
day, and  9  cents  on  Wednesday.     How  much  did  he  earn 
during  the  three  days?     How  many  are  35  and  29? 

9.  William  saw  two  flocks  of  wild  geese;  the  first  of  27 
geese,  the  second  of  23  (20  +  3).     How  many  geese  did  he 
see?     How  many  are  27  and  23? 

10.  Paid  9  cents  for  raisins,  15  cents  for  plums,  and  27 
(20  +  7)  cents  for  currants.     How  much  did  all  cost? 

11.  During  a  certain  recitation  29  questions  were  answered 
correctly  and    16    incorrectly.      How   many  questions   were 
asked?     How  many  are  29  and  10?     39  and  6? 

12.  Add  2  to  each  of  the  numbers  2,  12,  22,  32,  42,  etc., 
to  72. 

13.  Add  3  to  each  of  the  numbers  4,  14,  24,  etc.,  to  94. 

14.  Add  4  to  each  of  the  numbers  9,  19,  29,  39,  to  99. 


26  ADDITION. 

15.  Add  each  of  the  numbers  5,  6,  7,  8,  and  9  to  each  of 
the  numbers  6,  16,  26,  etc.,  to  96. 

16.  A  certain  school  had  40  girls  and  30  boys  in  attend- 
ance.    How  many  pupils  were  there  in  the  school? 

17.  A  music  teacher  paid  $12  for  a  metronome  and  $15 
for  music.     How  much  did  she  pay  for  both? 

18.  A  boy  bought  a  velocipede  for  $15  and  a  watch  for 
$20.     How  much  did  both  cost  him? 

19.  Mary  read  20  pages  of  history  one  day,  30  pages  the 
next,  and   25   the  next.      How  many  pages  did  she  read 
in  all? 

20.  In  a  certain  book-case  there  were  18   books  on  the 
upper  shelf,  20  on  the  next,  12  on  the  next,  and  10  on  the 
lowest.     How  many  books  in  the  case? 

21.  A  merchant  sold  15  yards  of  cloth  to  one  woman,  25 
to  another,  30  to  another,  and  25  to  another.     How  many 
yards  did  he  sell  to  them  all? 

22.  A  postmaster  sold  on  one  day  50  three-cent  stamps, 
65  on  another,  and  55  on  another.     How  many  stamps  did 
he  sell  in  the  three  days? 

23.  James  solved  31  oral  problems  and  24  written  prob- 
lems.    Harry  solved  35  oral  problems  and  25  written  prob- 
lems.    How  many  problems  did  each  solve?     How  many  did 
both  solve? 

24.  In  an  orchard  there  are  26  cherry-trees  and  31  apple- 
trees.     How  many  trees  are  there  in  the  orchard? 

25.  Henry  saw  three  flocks  of  wild  ducks,  the  first  con- 
taining 17  ducks,  the  second  25,  and  the  third  30.     How 
many  ducks  did  he  see? 

26.  James  paid  28  cents  for  a  slate,  20  cents  for  a  writing- 
book,  and  10  cents  for  ink.     How  much  did  he  pay  for  all? 

27.  The  month  of  January  has  31  days,  the  month  of  Feb- 
ruary has  28  days,  and  the  month  of  March  has  31  days. 
How  many  days  are  there  in  these  three  months? 


ADDITION.  27 

28.  How  many  acres  are  there  in  three  fields,  containing 
respectively  22  acres,  33  acres,  and  37  acres? 


WRITTEN    EXERCISES. 

53.    1.  What  is  the  sum  of  $535,  $213,  and  $384? 

PROCESS.        ANALYSIS. — For  convenience  we  arrange  the  numbers  to 
$  5  3  5      be  added  so  that  units  of  the  same  order  shall  stand  in  the 
same  column.     Beginning  with  the  lowest  order  of  units 
we  add  each  column  separately.    Thus,  4+3  +  5  =  12, 
the  sum  of  the  units.     12  units  are  equal  to  1  ten  and  2 


$  1 1 3  2  units.  We  write  2  under  the  column  of  units  and  reserve 
the  1  to  add  with  the  tens. 

1  reserved  +  8  +  1  +  3  =  1 3,  the  sum  of  the  tens.  13  tens  are 
equal  to  1  hundred  and  3  tens.  We  write  the  3  under  the  column 
of  tens  and  reserve  the  1  to  add  with  the  hundreds. 

1  reserved  +  3  +  2  +  5  =  11,  the  sum  of  the  hundreds.  11  hun- 
dreds are  equal  to  1  thousand  and  1  hundred,  which  we  write  in 
thousands'  and  hundreds'  places  in  the  sum. 

Hence  the  sum  is  $1132. 

1.  In  adding,  name  results  only.     Thus,  instead  of  saying  4  and  3 
are  7  and  5  are  12,  say  4 ,  7,  12. 

2.  When  the  sum  of  any  column  is  exactly  10,  20,  or  any  number 
of  tens,  we  write  0  under  the  column  added  and  reserve  the  1,  2,  3, 
etc.,  to  add  with  the  next  column. 

54.  RULE. — Arrange  the  numbers  so  that  units  of  the  same 
order  shall  stand  in  the  same  column. 

Begin  at  the  right,  add  each  column  separately,  and  unite  the 
sum,  if  it  is  less  than  ten,  under  the  column  added. 

If  the  sum  of  any  column  be  ten  or  more,  write  the  unit  figure 
only  under  that  column  and  add  the  ten  or  tens  with  the  next 
column. 

Write  the  entire  sum  of  the  last  column. 

PROOF. — Add  each  column  in  the  reverse  order.  If  the  re- 
sults agree,  the  work  is  probably  correct. 


28  ADDITION. 

EXAMPLES. 
Copy,  add,  and  prove: 

(2.)               (3.)             (4.)  (5.) 

310               512            $24.15  $12.25 

114               415              10.21  9.08 

523                371                8.34  7.15 


(6.)  (7.)  (8.)  (9.) 

POUNDS.  HORSES.  PLOWS.  RODS. 

4134  8104  3910  45 

2460  3673  418  3061 

3782  1856  1916  418 

469  7206  39  6 


10.  Add  4834,  3910,  4826,  8404. 

11.  Add  3159,  7816,  5459,  3568. 

12.  Add  $16.05,  $10.38,  $77.055. 

13.  Add  $317.50,  $600.10,  $514.085,  $6.16. 

14.  What  is  the  sum  of  thirty-six  thousand,  three  hundred 
five;  eight  hundred  ninety-seven  thousand,  nineteen? 

15.  What  is  the  sum  of  fifty-nine  thousand ;   seven  thou- 
sand, three  hundred  twelve;  sixty-eight  thousand,  four  hun- 
dred twenty-seven? 

16.  What  is  the  sum  of  three  hundred  forty-four  million, 
eight  hundred  ninety-six  thousand,  four  hundred  thirty-five; 
five  million,  six  thousand,  three;   forty-eight  thousand,  two 
hundred  ? 

17.  What  is  the  sum  of  eighteen  dollars,  five  cents;  fifty- 
one  dollars,  fifty-one  cents ;   ten  dollars,  ten  cents ;   eighteen 
dollars,  twenty-four  cents ;   thirty-five  dollars,  four  cents  ? 

18.  A  owns  345  sheep,  B  owns  295,  C  owns  436,  D  owns 
524.     How  many  sheep  do  all  own? 


ADDITION.  29 

19.  A  man  sold  his  piano  for  $413,  his  collection  of  paint- 
ings for  $536,  his  library  for  $719,  his  carpets  for  $728,  other 
furniture  for  $1,736,  his  horses,  carriage  and  two  sets  of  har- 
ness for  $1,324,  and  his  house  for  $9,137.    How  much  money 
did  he  obtain  by  the  sale? 

20.  A  fruit-dealer  shipped  for  New  York,   3,932  bushels 
of  apples  in  one  week,  2,436  in  the  next,  4,197  in  the  third, 
and,  within  the  next  month,   10,937  bushels.      What  was 
the  entire  number  of  bushels  shipped  by  him  during  that 
time? 

21.  A  man  making  his  will,  left  $3,450  to  his  wife,  $2,675 
to  his  oldest  son,  $1,850  to  his  second  son,  and  $1,290  to  his 
youngest  son.     What  amount  of  money  was  bequeathed  in 
his  will? 

22.  A  man  owns  five  horses.     The  first  is  worth  $225,  the 
second   $325,  the  third   $450,  the  fourth   as   much   as   the 
second  and   third,  and   the  fifth  as  much  as  the  first  and 
fourth.     What  is  the  value  of  the  five  horses? 

23.  A  and  B  were  building  a  brick  store.     The  first  day 
A  laid  2,136  bricks,  and  the  second  day  he  laid  as  many 
as  the  first  day  plus  207.     B,  on  the  first  day,  laid  1,936, 
and,  on  the  second  day>  341  more  than  on  the  first.     How 
many  bricks  were  laid  by  both  in  the  two  days? 

24.  The    distance    from    Greening    to    Chatfield    is    277 
miles,  from  Chatfield  to  Glendale  is  325  miles,  from  Glen- 
dale  to  Wyoming  is  139  miles,  from  Wyoming  to  Dale  is 
193  miles.     By  this  route  what  is  the  distance  from  Green- 
ing to  Dale? 

25.  A   man  took    2,126    steps   going   from    home   to  his 
place    of   business,    3,197   while  in  his  store,   6,239    going 
from   there  to   the   park,   5,782  while  in    the    park,   8,573 
going    from    there    home.      What   was    the   whole    number 
of  steps  taken  by  him  from  the  time  he  left  until  he  re- 
entered  his  house? 


30  ADDITION. 

26.  In  the  first  story  of  a  house,  the  hall  contained  117 
square  feet,  the  parlor  327,  the  sitting-room  296,  the  dining- 
room  257.     How  many  square  feet  of  carpeting  would   be 
required  to  cover  the  floors  of  these  rooms? 

27.  In  1870,  the  population  of  Buffalo  was  117,714;  that 
of  Eochester,  62,386;  that  of  Albany,  69,422;  that  of  Brook- 
lyn,  396,099.     How  many  inhabitants  did  these  four  cities 
contain? 

28.  The  area  of  Spain  is  195,773  square  miles;   that  of 
France,  204,091;  that  of  Switzerland,  15,922;  that  of  Italy, 
112,622.     Over  how  many  square  miles  do  the  four  countries 
extend  ? 

29.  A  speculator  bought  five  lots  for  $1,375  each.     He 
sold  the  first  for  $125  more  than  cost,  the  second  for  $319 
more  than  cost,  the  third  for  $291  more  than  cost,  the  fourth 
for  $739  more  than  cost,  and  the  fifth  for  $135  more  than 
cost.     How  much  money  did  he  receive  for  all? 

30.  The  State   of  Alabama   contains   1,430  libraries  and 
576,882  volumes;    Mississippi,   2,788  libraries  and  488,482 
volumes;   Louisiana,  2,332  libraries  and  847,406  volumes; 
Texas,  455  libraries  and  87,111  volumes.     How  many  libra- 
ries and  how  many  volumes  do  the  four  States  contain  ? 

31.  The  population  of  five  of  the  principal  cities  of  Ohio 
was   in   1870   as   follows:    Cincinnati,    216,239;    Cleveland, 
92,829;  Toledo,  31,584;  Columbus,  31,274;  Dayton,  30,473. 
What  was  the  entire  population  of  these  cities  in  1870? 

32.  The  population  of  five  of  the  principal  cities  of  Illinois 
was  in  1870  as  follows:   Chicago,  298,977;  Quincy,  24,025; 
Peoria,  22,849;    Springfield,  17,364;    Bloomington,  14,590. 
What  was  the  entire  population  of  these  cities  at  that  time? 

33.  In  1870,  the  population  of  St.  Louis,  Mo.,  was  310,864; 
Memphis,  Tenn.,  40,226;   Charleston,  S.  C.,  48,956;  Kich- 
mond,    Va.,    51,038;    New   Orleans,    La.,    191,418.     What 
was  the  entire  population  of  these  cities  at  that  time? 


ADDITION.  31 

34.  The  Warsaw  Manufacturing   Company  sawed   11,936 
feet  of  pine  on  Monday,  12,117  feet  of  hemlock  on  Tues- 
day, 8,135  feet  of  maple  on  Wednesday,  and  9,963  feet  of 
ash  on  Thursday.     How  many  feet  of  timber  did  they  saw 
in  the  four  days? 

35.  According  to  the  census  of  1870,  the  number  of  native 
Americans  in  Nebraska  was  92,245;    the  number  of  Irish, 
4,999;  of  Germans,  10,954;   of  English,  3,602;  of  Scotch, 
792;  of  Canadians,  2,632;  of  French,  340;   of  Norwegians, 
506;  of  Swedes,  2,352.    What  was  the  total  population  of  the 
State  in  1870? 

36.  In    a    certain    State    there   were    raised,    last    year, 
7,771,009   bushels  of  potatoes,  278,798  bushels   of  wheat, 
1,089,888    bushels   of   Indian    corn,    2,351,354   bushels    of 
oats,    658,816    bushels    of    barley.      What   was   the   entire 
number  of  bushels  of  farm  products  raised  that  year? 

37.  Mr.   George   Peabody  gave   to   the   poor  of  London 
$2,250,000,  to  the  town  of  Danvers  $60,000,  to  the  Grin- 
nell  Arctic  Expedition   $10,000,   to   the   city  of  Baltimore 
$1,000,000,  to  Phillips'    Academy   $25,000,    to   the    Massa- 
chusetts   Historical    Society   $20,000,    to   Harvard   College 
$150,000,    to   Yale    College    $150,000,    to    the    Southwest 
$1,500,000.      How    much    did    this    benevolent    gentleman 
give  away? 

38.  In  1870,  there  were,  in  the  United  States,  574  daily 
newspapers,  with  a  circulation  of  2,601,547;    107  tri-week- 
lies,  with  a  circulation  of  155,105;   115  semi- weeklies,  with 
a  circulation  of  247,197;   4295  weeklies,  with  a  circulation 
of   10,594,643;    96    semi-monthlies,   with    a    circulation    of 
1,349,820;   622  monthlies,  with  a  circulation  of  5,650,843; 
13  bi-monthlies,  with  a  circulation  of  31,650;    49  quarter- 
lies, with  a  circulation  of  211,670.     How  many  periodicals 
were  there  in  the  United  States  during  that  year,  and  what 
was  their  entire  circulation? 


32  ADDITION. 

39.  Mr.  A.  deposited  in  the  First  National  Bank  of  Albany, 
N.  Y.,  on  July  3,  1877,  $395.25;  on  July  5,  $874.75;  on 

July  8,  $325.85.  He  also  deposited  in  the  National  Park 
Bank  of  New  York  City,  on  July  12,  1877,  $1,546.87;  on 

July  16,  $1,275;  on  July  20,  $1,985.50.  How  much  did  he 
deposit  in  each  of  the  banks?  How  much  in  both  banks? 

(40.)  (41.)      (42.)      (43.) 

2134  6166      5873      46321 

8060  5878      3858      69788 

5032  9876      6430      76434 

8797  7977      5082      68924 

9888  6503      6353      96355 

6432  4556       4202      88789 

5421  6432      8792      93745 

(44.)  (45.)  (46.)  (47.) 

813  760  3945  5063 

976  500  9204  2050 

432  750  8769  3254 

397  694  9876  4200 

788  942  8020  6131 

643  293  5612  5945 

564  978  3424  2763 

'321  785  5861  4828 

156  696  2188  7688 

642  785  7654  3288 

321  688  3210  5634 

876  762  8765  6546 

543  451  5849  3250 

429  984  8574  7864 

386  579  9836  9758 

595  384  8759  8410 


INDUCTIVE    EXERCISES. 

55.    1.  If  I  have  6  peaches  and  give  away  3  of  them,  how 
many  will  be  left? 

2.  If  James  has  4  bunches  of  grapes  and  eats  2  of  them, 
how  many  will  be  left? 

3.  If  I  have  7  bunches  of  grapes  and  give  away  4  of 
them,  how  many  will  be  left? 

4.  If  you  find  8  acorns  and  lose  4  of  them,  how  many 
will  be  left? 

5.  How  many  are  left  when  4  things  are  taken  from  8 
things?     How  many  are  8  less  4?     7  less  4?     5  less  4?     9 
less  4?     6  less  4? 

6.  A  farmer  who  had  7  horses,  sold  3  of  them.     How 
many  had  he  left?     How  many  are  7  less  3?     9  less  3? 

7.  James  earned,  during  the  summer,  $9.     He  spent  $5 
of  the  money  for  a  coat,  and  the  rest  for  a  pair  of  boots. 
How  much  did  the  boots  cost  him? 

8.  Nine  is  how  many  more  than  5?     Than  6?     Than  4? 
Than  3? 

9.  A  boy  who  had  9  chickens,  sold  3  of  them.     How 
many  had  he  left? 

10.  Lawrence  had  10  pictures  in  his  room.     He  gave  his 
sister  3  of  them.     How  many  were  left  in  his  room? 

11.  A  man   earned  $11   per  week  and   spent  $7.     How 
much  did  he  save  weekly? 

(33) 


34  SUBTRACTION. 

12.  A  hen  had  nine  chickens,  but  5  of  them  were  killed. 
How  many  chickens  were  left?     How  many  must  be  added 
to  5  to  make  9? 

13.  When  5  is  taken  from  9,  what  number  is  left? 

14.  When   7   is  taken  from   10,   what  number  remains? 
How  many  must  be  added  to  7  to  equal  10? 

15.  Howard  is  10  years  of  age  and  Herbert  is  8.     What 
is  the  difference  in  their  ages?     What  is  the  difference  be- 
tween 10  and  8? 

16.  If  the  difference  between  10  and  8  be  added  to  8, 
what  will  the  result  be? 

17.  If  the  difference  between  any  two  numbers  be  added 
to  the  smaller  number,  to  what  will  the  result  be  equal? 

18.  What  is  the  difference  between  6  horses  and  4  horses? 
Between  6  horses  and  5  cents? 

19.  Why  can  you  not  find  the  difference  between  6  horses 
and  5  cents? 

20.  Why  can  you  find  the  difference  between  6  horses  and 
4  horses? 

21.  Between  what  kinds  of  numbers  only  can  the  differ- 
ence be  found? 

DEFINITIONS. 

56.  Subtraction  is  the  process  of  taking  one  number 
from  another. 

57.  The  Minuend  is  the  number  from  which  another 
is  to  be  subtracted. 

58.  The  Subtrahend  is  the  number  to  be  subtracted. 

59.  The  Remainder,  or  Difference,  is  the  result 
obtained  by  subtracting. 

60.  The  Sign  of  Subtraction  is  a  short  horizontal 
line:  — .     It  is  named  minus. 


SUBTRACTION. 


35 


When  the  sign  minus  is  placed  between  two  numbers  it 
shows  that  the  one  after  it  Is  to  be  subtracted  from  the  one 
before  it. 

Thus,  9  —  5  is  read  9  minus  5,  and  means  that  5  is  to  be  sub- 
tracted from  9. 

61.  PRINCIPLE. — 1.   Only  like  numbers  can  be  subtracted. 
2.    The  sum  of  the  subtrahend  and  remainder  must  be  equal  to 
the  minuend. 

TABLE. 


1—1=  0 

2—2=  OJ   3—3=  0 

4—4=  0|   5—  5=  0 

2—1=  1 

3—2=  1 

4—3=  1 

5—4=  1 

6—  5=  1 

3—1=  2 

4—2=  2 

5—3=  2 

6—4=  2 

7—  5=  2 

4—1=  3 

5—2=  3 

6—3=  3 

7—4=  3 

8—  5=  3 

5—1=  4 

6—2=  4 

7—3=  4 

8—4=  4 

9—  5=  4 

6—1=  5 

7—2=  5 

8—3=  5 

9—4=  5 

10—  5=  5 

7—1=  6 

8—2=  6 

9—3=  6 

10—4=  6 

11—  5=  6 

8—1=  7 

9—2=  7 

10—3=  7 

11—4=  7 

12—  5=  7 

9—1=  8 

10—2=  8 

11—3=  8 

12—4=  8 

13—  5=  8 

10—1=  9 

11—2=  9 

12—3=  9 

13—4=  9 

14—  .5=  9 

11—1=10 

12—2=10 

13-3=10 

14—4=10 

15—5=10 

6—6=  0 

7—7=  0 

8-8=  0 

9—9=  0 

10—10=  0 

7—6=  1 

8—7=  1 

9—8=  1 

10—9=  1 

11—10=  1 

8-6=  2 

9—7=  2 

10—8=  2 

11—9=  2 

12—10=  2 

9—6=  3 

10—7=  3 

11—8=  3 

12—9=  3 

13—10=  3 

10-6=  4 

11—7=  4 

12-8=  4 

13—9=  4 

14—10=  4 

11—6=  5 

12—7=  5 

13—8=  5 

14—9=  5 

15—10=  5 

12—6=  6 

13—7=  6 

14—8=  6 

15—9=  6 

16—10=  6 

13—6=  7 

14—7=  7 

15—8=  7 

16—9=  7 

17—10=  7 

14—6=  8 

15—7=  8 

16—8=  8 

17—9=  8 

18—10=  8 

15—6=  9 

16—7=  9 

17—8=  9 

18—9=  9 

19—10=  9 

16—6=10 

17—7=10 

18—8=10 

19—9=10 

20—10=10 

36  SUBTRACTION. 

CASE     I. 

62.  When  no  figure  of  the  subtrahend  has  a  greater 
value  than  the  corresponding  figure  of  the  minuend. 

1.  A  merchant  had  15  barrels  of  flour,  and  sold  4  of 
them.     How  many  had  he  left? 

ANALYSIS. — Since  he  had  15  barrels  of  flour  and  sold  4  of  them,  he 
had  left  the  difference  between  15  barrels  and  4  barrels,  which  is  11 
barrels. 

2.  Alice  bought  18  cakes,  and  ate  6  of  them.    How  many 
had  she  left? 

3.  James  saw  17  birds  on  a  tree,  but  7  soon  flew  away. 
How  many  remained? 

4.  If  a  man  earns  $19  a  week,  and  spends  $9,  how  much 
will  he  save  each  week? 

5.  Lewis  owed  his  brother  $7,  and  paid  him  S3.     How 
much  did  he  still  owe  him? 

6.  Eliza  had  16  plums,  but  gave  5  to  her  father.     How 
many  had  she  left? 

7.  James  had   $12,   and  lost  $2.      How   many  had  he 
left? 

8.  If  John  is  19  years  old,  and  Maggie  13,  how  much 
younger  than  John  is  Maggie? 

9.  In  the  same  shop  6  boys  and  17  men  work.     How 
many  more  men  than  boys  are  there  in  the  shop? 

10.  There  were   18  girls  and   7   boys  in   a  class.     How 
many  more  girls  than  boys  were  there? 

11.  Laura  had  14  cents,   and  lost  3  cents.     How  many 
had  she  then? 

12.  Henry   solved   19   examples,    and   George    solved    8. 
How  many  more  did  Henry  solve  than  George? 

130  William  wrote  16  lines  in  his  copy-book,  and  Peter 
wrote  5  lines  less.  How  many  did  Peter  write? 


SUBTRACTION.  37 


14.  One  piece  of  cloth  contained  20  yards  and  another  10 
yards.     How  many  yards  more  were  there  in  the  larger  piece  ? 

15.  A  boy  had  24  chickens,  and  10  of  them  died.     How 
many  had  he  left? 

16.  Julia  gave  me  11  cents."    If  she  had  16  cents  at  first, 
how  many  had  she  left? 

17.  A  girl  bought  18  eggs,  and,  on  her  way  home,  fell  and 
broke  5  of  them.     How  many  had  she  left? 

18.  Subtract  by  2's  from  22  to  0;    thus:  22,  20,  18,  16, 
14,  12,  etc. 

19.  Subtract  by  3's  from  35  to  2.     From  45  to  0. 

20.  Subtract  by  4's  from  48  to  0.     From  45  to  1. 

21.  Count  back  by  5's  from  35  to  0.     From  59  to  4. 


WRITTEN    EXERCISES. 

63.    1.  From  547  subtract  235. 

PROCESS.  ANALYSIS. — For  convenience  we  write  the  less 

Minuend       547      number  under  the  greater,  units  under  units,  tens 
Subtrahend   235      un^er  tens>  etc->  an(*  subtract  each  order  of  units 

separately  from  the  same  order  of  the  minuend. 

Remainder    312  Thus,    7    units  —  5    units  =  2  units,  which  we 

write  under  the  units. 

4  tens  —  3  tens  =  1  ten,  which  we  write  under  the  tens. 

5  hundreds  —  2  hundreds  =  3  hundreds,  which  we  write  under  the 
hundreds. 

Hence  the  remainder  is  312. 

PROOF. — 312,  the  remainder,  plus  235,  the  subtrahend,  equals  547, 
the  minuend.     Hence  the  result  is  correct. 

Copy,  subtract,  and  prove: 

(2.)  (3.)  (4.)  (5.)  (6.)  (7.) 
713  458  986  854  795  7842 
302  134  732  641  433  2310 


38  SUBTRACTION. 


(8.) 

(9.) 

(10.) 

(11.) 

(12.) 

$48.25 

$64.29 

$45.78 

$38.94 

$41.89 

23.13 

30.29 

34.65 

27.83 

20.45 

13.  A  drover,  having   1583   sheep,  sold   1441   of  them. 
How  many  had  he  left? 

14.  A  speculator  bought  some  land  for  $5849.75,  and  sold 
it  for  $6959.95.     How  much  did  he  gain? 

15.  A  cotton  factory  made   9875   yards  of  cloth  in  one 
week,  and  sold,  during  the  same  time,  7652  yards.     How 
much  more  was  made  than  sold? 

16.  A   money-lender   received   for  interest,   during   1875, 
$1685.49,  and  during  1876,  $2796.59.     In  which  year  did 
he  receive  the  greater  sum,  and  how  much? 

17.  A  man  bought  7467  bricks,  and  carted  away  3136. 
How  many  remained  to  be  moved? 

18.  A  has  736  sheep,  and  B  has  213  less  than  A.     How 
many  sheep  has  B? 

19.  A   man  gave   his  note  for  $6792,  without   interest. 
In  two  years  he  had  paid  $3401.     How  much  did  he  still 
owe  on  the  note? 

20.  A  man  bought  a  house  for  $1765,  and  afterward  sold 
it,  thereby  losing  $504.     For  how  much  did  he  sell  it? 

21.  A  man  bought  a  span  of  horses  for  $364,  and  a  yoke 
of  oxen  for  $120.     How  much   more  did  he  give  for  the 
horses  than  for  the  oxen? 

22.  A  merchant  having  6755  yards  of  cloth,  sold  2532 
yards.     How  many  yards  had  he  remaining? 

23.  A  father  having  3652  acres  of  land,  gave  his  son  1230 
acres.     How  many  acres  had  he  left? 

24.  A  vintner  had  38756  gallons  of  wine,  and  sold  during 
the  year,  34243  gallons.     How  much  remained  unsold? 

25.  The  circulation   of  a  newspaper  in   1875  was  38293 
copies,  and  in  1876,  37180.     What  was  the  decrease? 


SUBTRACTION.  39 


CASE  II. 

64.  When  any  figure  of  the  subtrahend  has  a  greater 
value  than  the  corresponding  figure  of  the  minuend. 

1.  A  gentleman  bought  a  coat  at  $40,  and  a  vest  at  $9; 
he  gave  the   merchant  a  hundred-dollar  bill.     How  much 
change  ought  he  to  receive? 

ANALYSIS. — Since  he  paid  $40  for  a  coat  and  $9  for  a  vest,  for  both 
he  paid  the  sum  of  $40  and  $9,  or  $49.  And  since  he  gave  the  mer- 
chant $100,  he  should  receive  the  difference  between  $100  and  $49. 

4100— $40=$GO;  $60— $9  =  $51.    Therefore  he  should  receive  $51. 

2.  A  boy  saw  15  birds  on  a  tree,  and  9  of  them  flew  away. 
How  many  remained? 

3.  John  is  16  years  old,  and  James  is  8.    How  much  older 
than  James  is  John? 

4.  A  jeweler  bought  a  watch  for  $75.  and  sold  it  for  $100. 
How  much  did  he  gain  by  the  operation? 

5.  A  grocer  bought  a  quantity  of  sugar  for  $36,  and  re- 
tailed the  same  for  $50.     How  much  did  he  gain  by  the  sale? 

6.  A  boy  had  34  marbles,  and  gave  away  9  of  them. 
How  many  had  he  left? 

7.  A  lady  bought  a  chair  for  $3,  and  a  table  for  $5;  she 
gave  a  twenty-dollar  bill  to  the  cabinet-maker.,    How  much 
change  ought  she  to  receive? 

8.  A  man  set  out  to  walk  50  miles;  he  walked  20  miles 
the  first  day,  and  19  the  second  day.     How  many  miles  were 
left  for  him  to  walk? 

9.  A  man  bought  a  cow  for  $35,  and  sold  her  for  $43, 
after  keeping  her  4  weeks  at  an   expense  of  $2  per  week. 
How  much  did  he  gain? 

10.  A  man  who  earned  $60  a  month,  paid  $25  a  month  for 
his  board,  and  $15  a  month  for  other  expenses.  How  much 
did  he  save  per  month? 


40  SUBTRACTION. 

11.  Count  back  by  10's  from  107  to  7;  thus:  107,  97,  etc. 

12.  Count  back  by  10's  from  95  to  5.     From  79  to  9. 

13.  Count  back  by  10's  from  83  to  13.     From  98  to  18. 

14.  Subtract  by  20's  from  106  to  26;  thus:  106,  86,  etc. 

WRITTEN    EXERCISES. 

65.  1.  From  643  subtract  456. 

PROCESS.  ANALYSIS. — We  write  the  numbers  as  in  the  previous 

g  4  g  case  and  begin  at  the  right  to  subtract. 

Since  6  units  can  not  be  subtracted  from  3  units,  we 
unite  with  the  3  units  a  unit  of  the  next  higher  order, 
187  which  is  equal  to  10  units,  making  13  units:  6  units  from 

13  units  leave  7  units,  which  we  write  under  the  units. 
Since  one  of  tens  was  united  with  the  units,  there  can  be  but  3  tens 
left.  Because  5  tens  can  not  be  subtracted  from  3  tens,  we  unite  with 
the  3  tens  as  before,  a  unit  of  the  next  higher  order,  which  is  equal 
to  10  tens,  making  13  tens:  5  tens  from  13  tens  leave  8  tens, 
which  we  write  under  the  tens. 

Since  one  of  the  hundreds  was  united  with  the  tens,  there  are  but  5 
hundreds  left:  4  hundreds  from  5  hundreds  leave  1  hundred,  which 
we  write  under  the  hundreds.  Hence  the  result  is  187. 

PROOF. — 187,  the  remainder,  plus  456,  the  subtrahend,  equals  643? 
the  minuend.  Hence  the  result  is  correct.  • 

66.  RULE. — Write  the  subtrahend  under  the  minuend,  units 
under  units,  tens  under  tens,  ete. 

Begin  at  the  right  and  subtract  each  figure  of  the  subtrahend 
from  the  corresponding  figure  of  the  minuend,  writing  the  result 
beneath. 

If  a  figure  in  the  minuend  has  a  less  value  than  the  corre- 
sponding figure  in  the  subtrahend,  increase  the  former  by  ten,  and 
subtract ;  then  diminish  by  one,  the  units  of  the  next  higher  order  in 
the  minuend,  and  subtract  as  before. 

PROOF. — Add  together  the  remainder  and  subtrahend.  If  the 
result  be  equal  to  the  minuend  the  work  is  correct. 


SUBTRACTION. 


41 


EXAMPLES. 
Copy,  subtract,  and  prove: 

(2.)  (3.)  (4.)  (5.)  (6.) 

753  984  826  754  1426 

448  756  534  482  547 


(7.) 

(8.) 

(9.) 

(10.) 

(11.) 

843 

1846 

1683 

2897 

3001 

782 

927 

1395 

1598 

2851 

(12.) 

(13.) 

(14.) 

(15.) 

(16.) 

$24.45 

$39.18 

$63.25 

$71.89 

$42.34 

21.38 

27.92 

47.18 

47.93 

18.67 

Find  the  difference  between 

17.  583  and  294. 

18.  690  and  508. 

19.  763  and  574. 

20.  966  and  599. 

21.  982  and  796. 

22.  891  and  798. 

23.  5833  and  4968. 

24.  7521  and  3635. 


25. 

26. 

27. 
28. 
29. 
30. 
31. 


7812  and 

8003  and 

63004  and 

65432  and 

69721  and 

78303  and 

865932  and 


32.  9050308  and 


1984. 

5872. 

54872 

54862. 

49653. 

49424. 

785841. 

563420. 


33.  A  man  set  out  on  a  journey  of  861  miles.     During  the 
first  day  he  traveled  297  miles,  and  during  the  second  day 
308  miles.     How  many  miles  had  he  yet  to  travel? 

34.  A  merchant  deposited  in  a  bank  on  Monday  $584;  on 
Tuesday,  $759;   on  Wednesday,  $463.     He  drew  out  $1298 
during  that  time.     How  much  did  his  deposits  exceed  what 
he  drew  out? 


42  SUBTRACTION. 

35.  A  grocer  had  3715  pounds  of  sugar  on  hand.     On  one 
day  he  sold  1235  pounds,  on  the  next  1317;   the  third  day 
he  sold  to  C  all  the  sugar  that  remained.     How  many  pounds 
did  C  buy? 

36.  I  bought  a  horse  for  $637,  and  a  cow  for  $317.     I  sold 
the  horse  for  $729,  and  the  cow  for  $356.     How  much  did 
I  gain  by  the  sale? 

37.  In  the  first  of  three  pavements  there  are  1317  bricks, 
in  the  second  there  are  2357,  in  the  third  there  are  1719 
less  than  in  both  the  others.     How  many  bricks  in  the  third 
pavement? 

38.  In  1869  there  were  264,146,900  bushels  of  wheat  raised 
in  the  United  States,  and  874,120,005  bushels  of  corn.     How 
much  more  corn  than  wheat  was  produced? 

39.  A  bought  351  acres  of  land,  and  B  bought  27  acres 
more  than  A;  B  sold  his  land  to  C,  who  then  had  537  acres. 
How  many  acres  did  C  have  at  first? 

40.  A  grocer  retailed  a  quantity  of  sugar  for  $308. 40,  and 
so  gained  $106.28.     How  much  had  he  paid  for  it? 

41.  The  year  1870  was  just  378  years  after  the  discovery 
of  America  by  Columbus.     In  what  year  did  that  event  take 
place  ? 

42.  On  Monday  morning  a  bank  had  on  hand  $1826.     Dur- 
ing the  day  $2191  were  deposited  and  $3412  drawn  out;  on 
Tuesday  $3256  were  deposited  and  $2164  drawn  out.     How 
many  dollars  were  on  hand  Wednesday  morning? 

43.  E.  S.  Hill  is  worth  $15795,  of  which   $2895   is   in- 
vested in  bank  stock,  $3864  in   mortgages  and   the  rest  in 
land.     How  much  has  he  invested  in  land? 

44.  Of  the  two   numbers    89346   and   56849,  how  much 
nearer  is  the  one  than  the  other  to  68754? 

45.  The  number  of  pupils  who  attended  school  in  Boston 
in  1870  was  38944,  and  of  this  number  35442  attended  the 
public  schools.     How  many  attended  the  other  schools? 


MULTIPLICATION 


INDUCTIVE    EXERCISES. 

67.    1.  How  many  books  are  there  in  2  piles  containing  3 
books  each? 

2.  If  you  place  4  apples  in  a  group,  how  many  apples  are 
there  in  3  such  groups?     In  4  groups? 

3.  When  there  are  3  roses  in  a  cluster,  how  many  are 
there  in  3  clusters?     In  4  clusters?     In  5  clusters? 

4.  How  many  are  3  +  3  +  3  +  3,  or  four  3's? 

5.  How  many  are  4  +  4  +  4,  or  three  4's? 

6.  How  many  are  four  4's?     Four  5's?     Four  6's? 

7.  James  bought  5  pencils  at  5  cents  each.     How  much 
did  they  cost  him?     How  many  cents  are  5  times  5  cents? 
How  many  are  five  5's? 

8.  An  orchard  contains  5  rows  of  6  trees  each.     How 
many  trees  are  there  in  the  orchard?     How  many  trees  are 
5  times  6  trees?     How  many  are  5  times  6? 

9.  James  piled  his  blocks  in  3  piles,  each  containing  5 
blocks.     How  many  blocks  had  he?     How  many  are  3  times 
5  blocks?     How  many  are  3  times  5? 

10.  A  boy  earned  $4   a  week  for  6    weeks.     How  much 
did  he  earn  in  all?     How   many  dollars   are   6   times   $4? 
How  many  are  6  times  4? 

11.  Harry  played  5  hours  per  day.     How  many  hours  did 
he  play  in  6  days?     How  many  are  6  times  5  hours?     How 
many  are  6  times  5? 

(43) 


44  MULTIPLICATION. 

12.  How  does  5  times  4  compare  with  4  times  5?     5  times 
6  with  6  times  5? 

13.  When  numbers  are  used  without  reference  to  any  par- 
ticular thing,  they  are  called  Abstract  Numbers. 


DEFINITIONS. 

68.  Multiplication  is  a  short  process  of  finding  the 
sum  of  equal  numbers;   or, 

The  process  of  repeating  one  number  as  many  times  as 
there  are  units  in  another. 

69.  The  Multiplicand  is  the  number  to  be  repeated 
or  multiplied. 

70.  The  Multiplier  is  the  number  showing  how  many 
times  the  multiplicand  is  to  be  repeated. 

71.  The  Product  is  the  result  obtained  by  multiplying. 

72.  The  multiplicand  and  multiplier  are  called  the  factors 
of  the  product. 

73.  The   Sign   of  Multiplication  is   an  oblique' 
cross :   X  •     It  is  read,  multiplied  by,  or  times.     When  placed 
between  two  numbers  it  shows  that  they  are  to  be  multiplied 
together. 

Thus,  4  X  ^  is  read,  4  multiplied  by  3,  or  3  times  4. 

74.  PRINCIPLES. — 1.   The  multiplier  must  be  regarded  as  an 
abstract  number. 

2.  The  multiplicand  and  product  must  be  like  numbers. 

3.  Either  of  the  factors  may  be  used  as  multiplicand  or  multi- 
plier when  both  are  abstract. 

In  practice,  for  convenience,  the  smaller  number  is  generally  used 
as  multiplier. 


MULTIPLICATION.  47 

24.  If  6  men  can  do  a  piece  of  work  in  21  days,  how 
long  will  it  take  one  man  to  do  the  same  work? 

25.  In  a  certain  orchard  there  are  9  rows  of  trees  and  27 
trees  in  a  row.     How  many  trees  are  there  in  the  orchard? 

26.  Count  by  2's  from  0  to  24;  thus:  2,  4,  6,  8,  10,  etc. 

27.  Count  by  3's  from  0  to  36.     By  4's  from  0  to  48. 

28.  Repeat  all  the  numbers  of  times  5  from  once  5  to  10 
times  5.     Thus,  once  5  is  5,  2  times  5  are  10,  3  times  5  are 
15,  etc. 

29.  Repeat  from  once  6  to  10  times  6,  and  back  from  10 
times  6  to  once  6. 

30.  Repeat  from  once    7  to  10  times    7,  and  reverse. 

31.  Repeat  from  once    8  to  10  times    8,  and  reverse. 

32.  Repeat  from  once    9  to  10  times    9,  and  reverse. 

33.  Repeat  from  once  10  to  10  times  10,  and  reverse. 

34.  At  25  cents  a  pound,  how  much  will  6  pounds  of 
raisins  cost? 

35.  If  a  man  can  dig  28  bushels  of  potatoes  in  one  day, 
how  many  can  he  dig  in  4  days? 

36.  If  a  person  spend  25  cents  a  day  for  cigars,  how  much 
will  he  spend  in  7  days? 

37.  If  a  boy  earns  33  cents  a  day,  how  much  will  he  earn 
in  9  days? 

38.  When  butter  is  selling  at  37  cents  a  pound,  what  will 
7  pounds  cost  me? 

WRITTEN    EXERCISES. 

76.  1.  How  many  are  3  times  275? 

IST  PROCESS.  ANALYSIS. — Since   multiplication   is   a   short 

275  process  of  adding  equal  numbers,  it  is  evident 

275  that  we  can  determine  by  addition  how  many  3 

2  rj  p.  times  275,  or  three  275's,  are.    The  sum  is  825. 

In  practice,    a  shorter   method   is  employed, 
Sum  825  which  is  given  in  the  second  process  and  analysis. 


48 


MULTIPLICATION. 


2D  PROCESS. 

Multiplicand  275 
Multiplier  3 

Product  825 


ANALYSIS. — For  convenience  we  write  the 
multiplier  under  the  multiplicand,  and  begin 
at  the  right  to  multiply.  Thus,  3  times  5  units 
are  15  units,  or  1  ten  and  5  units.  We  write 
the  5  units  in  units'  place  in  the  product  and 
reserve  the  tens  to  add  with  the  tens. 

3  times  7  tens  are  21  tens,  plus  1  ten  reserved  are  22  tens,  or  2 
hundreds  and  2  tens.  We  write  the  2  tens  in  tens'  place  in  the 
product  and  reserve  the  hundreds  to  add  with  the  hundreds. 

3  times  2  hundreds  are  6  hundreds,  plus  2  hundreds  reserved  are  8 
hundreds,  which  we  write  in  hundreds'  place  in  the  product.  Hence 
the  product  is  825,  the  same  as  the  sum  in  the  first  process. 

PROOF. — If  the  results  obtained  by  both  processes  agree,  the  work 
is  probably  correct. 

In  multiplying,  pronounce  the  results  only.  Thus,  in  the  example 
given  above,  instead  of  saying  3  times  5  are  15,  3  times  7  are  21,  plus 
1  reserved  are  22;  3  times  2  are  6,  plus  2  reserved  are  8;  say  15, 
22,  8. 


Solve  and  prove: 

2.  3  times  314. 

3.  4  times  568. 

How  many  are 

8.  5  times  314? 

9.  4  times  655? 
10.  7  times  764? 


4.  4  times  987. 

5.  5  times  345. 


11.  3  times  830? 

12.  6  times  734? 

13.  9  times  $48? 


6.  5  times  $819. 

7.  3  times  $769. 


14.  8  times  $42? 

15.  6  times  $32? 

16.  7  times  $57? 


17.  If  a  man  earns  $17.25  per  week,  how  much  can  he 
earn  in  8  weeks? 

18.  A  benevolent  man  paid  annually  for  the  support  of 
the  poor  $2365.     How  much  did  he  pay  in  7  years? 

19.  A  shoe  dealer  sold  9  pairs  of  shoes  at  $3.75  a  pair. 
How  much  did  he  receive  for  all? 

20.  A  man  bought  8  cows  at  an  average  price  of  $31.27. 
How  much  did  they  all  cost  him? 


MULTIPLICATION,  49 

21.  If  a  ship  sail  425  miles  in  one  week,  how  far  will  she 
sail  in  9  weeks? 

22.  A  barrel  of  flour  weighs  196  pounds.     How  much  will 
8  barrels  weigh? 

23.  When  wheat  is  worth  $1.78  per  bushel,  how  much 
can  be  realized  from  the  sale  of  9  bushels? 

24.  At  $6.25  a  pair,  what  will  be  the  cost  of  7  pairs  of 
boots? 

25.  There  are  5280  feet  in  a  mile.     How  many  feet  in  7 
miles? 

26.  At  $37.50  an  acre,  what  will  be  the  cost  of  8  acres  of 
land? 

27.  What  will  be  the  cost  of  7  thousand  feet  of  lumber  at 
$18.25  per  thousand? 

28.  When  broom  corn  is  selling  at  $83.50  a  ton  what  is 
the  value  of  8  tons? 

CASE  II. 

77.  When  the  multiplier  is  expressed  by  more  than 
one  figure. 

1.  There  are  9  square  feet  in  one  square  yard.     How 
many  are  there  in  10  square  yards? 

2.  How  many  square  feet  in  6  square  yards? 

3.  Since  10  square  yards  contain  90  square  feet,  and  6 
square  yards  contain  54  square  feet,  how  may  the  number 
of  square  yards  in  10  -f  6,  or  16,   square  feet,   be  found? 
How,  then,  may  you  multiply  by  16?     By  18?     By  13? 

4.  Find  the  cost  of  17  yards  of  cloth  at  18  cents  a  yard? 

5.  When  eggs  are  21  cents  a  dozen,  what  will  15  dozen 
cost? 

6.  Since  12  inches  make  one  foot  in  length,  how  many 
inches  are  there  in  18  feet? 

7.  A  pound  of  sugar  is  equal  to  16  ounces.     How  many 
ounces  are  there  in  a  quantity  of  sugar  weighing  16  pounds? 


50  MULTIPLICATION. 

7.  Find  the  cost  of  17  yards  of  cloth  at  8  cents  a  yard, 
by  finding  the  cost  of  9  yards,  and  then  of  8  yards.     Of  10 
yards  and  7  yards. 

8.  What  will  be  the  cost  of  11  primers  at  25  cents  each? 

9.  Find  the  cost  of  16  yards  of  cloth  at  8  cents  a  yard, 
by  finding  the  cost  of  10  yards  and  6  yards.     9  yards  and 
7  yards.     8  yards  and  8  yards. 

10.  James  is  in  school  5  hours  a  day.     How  many  hours 
is  he  in  school  during  three  weeks,  or  15  school-days? 

11.  A  bought  4  sets  of  forks,  each  set  containing  6  forks. 
How  much  did  the  forks  cost  him  at  $2  each  ? 

12.  Mary  bought  15  pounds  of  sugar  at  11  cents  a  pound, 
and  3  pounds  of  raisins  at  15  cents  a  pound.     After  paying 
her  bill  she  had  10  cents  left.     How  much  money  had  she 
at  first? 

13.  A  cooper  can  make  12  barrels  a  day.     How  many  can 
he  make  in  12  days? 

14.  John  bought  12  lead  pencils  at  8  cents  each,  and  2 
erasers  at  4  cents  each.     How  much  did  all  cost  him? 

15.  The  railroad  fare  from  Rochester  to  New  York  is  $7. 
How  much  will  the  tickets  for  a  party  of  9  cost? 

16.  If  a  cow  give  9  quarts  of  milk  a  day,  how  much  milk 
will  she  give  in  9  days? 

17.  If  a  man  put  $8  in  a  savings-bank  each  month,  how 
much  will  he  deposit  in  a  year? 

18.  At  $4  a  yard,  what  will  17  yards  of  broadcloth  cost? 

19.  If  a  laborer  can  earn  $2  a  day,  how  much  can  he  earn 
in  12  days? 

20.  What  will  15  pairs  of  skates  cost  at  $4  a  pair? 

21.  At  20  cents  a  dozen,  how  much  will  18  dozen  eggs  cost? 

22.  A  coal  dealer  sold  an  average  of  18  tons  of  coal  per  day 
for  12  days.     How  many  tons  did  he  sell  in  that  time? 

23.  At  22  cents  a  pound,  how  much  will  11  pounds  of 
butter  cost? 


MULTIPLICATION. 


51 


24.  How  far  will  a  man  travel  in  15  days,  if  he  travel 
10  hours  a  day  and  3  miles  an  hour? 

25.  A  man  bought  25  cows  and  12  times  as  many  sheep. 
How  many  sheep  did  he  buy? 


WRITTEN     EXER  CISES. 


78.  1.  Multiply  327  by  123. 

1ST  PROCESS. 

327 
123 


ANALYSIS. — For  convenience  we  write 
the  numbers  as  in  the  preceding  case. 
Since  in  multiplying  we  must  multiply 
by  the  parts  of  the  multiplier  and  add 
the  partial  products,  to  multiply  by  123 
we  multiply  by  3  units,  2  tens,  and  1 
hundred  as  partial  multipliers. 

3  times  327  are  981,  the  first  partial 
product;  2  times  327  are  654  and  2  tens 
times  327  are  654  tens,  or  6540,  which 
we  write  for  a  second  partial  product. 

1  time  327  equals  327,  and  1  hundred  times  327  are  327  hundreds, 
or  32700,  which  we  write  for  a  third  partial  product.  The  sum  of 
the  partial  products  will  be  the  entire  product. 


1st  Partial  Prod. 
2d  Partial  Prod. 
3d  Partial  Prod. 


981 

6540 

32700 


Entire  Prod.          40221 


1st  Partial  Prod. 
2d  Partial  Prod. 
3d  Partial  Prod. 


2D   PROCESS. 

327 
123 

981 
654 
327 


ANALYSIS. — In  the  second  process  the 
ciphers  at  the  right  of  the  partial  prod- 
ucts are  omitted,  the  significant  figures 
still  occupying  their  proper  places.  Thus, 
in  multiplying  by  2  tens  the  product  was 
654  tens,  or  6  thousand,  5  hundred,  4 
tens,  which  we  write  in  their  places  in 
the  partial  product. 

In  multiplying  by  hundreds,  the  low- 
est order  of  the  product  is  hundreds, 
hence  we  write  the  first  figure  of  the  product  under  hundreds. 

PROOF. — Multiply  the  multiplier  by  the  multiplicand.  (Prin.  3.) 
If  the  result  agrees  with  that  formerly  obtained,  the  work  is  probably 
correct. 


Entire  Prod.          40221 


52 


MULTIPLICATION. 


RULE. —  Write  the  multiplier  under  the  multiplicand  with 
units  under  units,  tens  under  tens,  etc. 

Multiply  each  figure  of  the  multiplicand  by  each  significant  fig- 
ure of  the  multiplier  successively,  beginning  with  units.  Place  the 
right  hand  figure  of  each  product  under  the  figure  of  the  multiplier 
used  to  obtain  it,  and  add  the  partial  products. 

PROOF. — Review  the  work,  or  multiply  the  multiplier  by  the 
multiplicand.  If  the  results  agree  the  work  is  probably  correct. 


EXAMPLES. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

Multiply  325 

219 

384 

$2.81 

$3.18 

By             42 

54 

46 

23 

36 

Multiply 

: 

Multiply  : 

7. 

456  by  12. 

23.  73982  by  321. 

8. 

389  by  23. 

24.  42586  by  604. 

9. 

493  by  25. 

25.  89258  by  703. 

10. 

374  by  27. 

26.  84206  by  569. 

11. 

3625  by  28, 

27.  156783  by  423. 

12. 

2413  by  31. 

28.  248164  by  372. 

13. 

3681  by  63. 

29.  182642  by  419. 

14. 

67021  by  52. 

30.  192573  by  429. 

15. 

63583  by  62. 

31.  234567  by  612. 

16. 

84216  by  78. 

32.  467105  by  623. 

17. 

38413  by  35. 

33.  398120  by  706. 

18. 

29615  by  45. 

34.  683912  by  1684. 

19. 

23423  by  25. 

35.  312465  by  1827. 

20. 

24542  by  64. 

36.  468975  by  2946. 

21. 

45684  by  73. 

37.  416004  by  3009. 

22. 

41075  by  62. 

38.  329706  by  3802. 

MULTIPLICATION.  53 


39.  $  18.61  by  73. 

40.  $115.81  by  45. 

41.  $164.32  by  81. 

42.  $123.45  by  804. 

43.  $415.05  by  367. 


44.  $  18.37  by  127. 

45.  $113.41  by  613. 

46.  $281.69  by  247. 

47.  $312.09  by  684. 

48.  $425.27  by  618. 


49.  In  a  reaper  factory  an  average  of  2346  reapers  is  con: 
structed  annually.     At  this  rate  how  many  would  be  made 
in  25  years? 

50.  A  farmer  counted  the  trees  in  his  orchard  and  found 
that  he  had  104  rows,  each  row  containing  106  trees.     How 
many  trees  were  there  in  the  orchard? 

51.  In   a   croquet   factory  a  man  makes  835  balls  daily. 
How  many  balls  can  he  make  in  312  days? 

52.  The  distance  between  Rochester  and  Syracuse  is  81 
miles.     How  many  miles  per  month  of  31  days,  will  a  loco- 
motive travel  that  goes  from  Rochester  to  Syracuse  daily  and 
returns? 

53.  Mr.  Davis  built  8  houses  at  a  cost  of  $1925  each 
6  at  $2275  each,  and  5  at  $3897  each.     What  did  they  all 
cost  him? 

54.  Sold  my  farm  of  413  acres  at  $85  per  acre.     How 
much  did  I  get  for  it? 

55.  How  much  will  it  cost  to  build  89  miles  of  railroad 
at  an  estimated  expense  of  $57394  per  mile? 

CASE  III. 

79.  Where  there  are  ciphers  on  the  right  of  either 
or  hoth  factors. 

1.  How  many  are  are  10  times  2?     3?     4?     5?     6?     7? 
8?     9? 

2.  Write  the  above  multiplicands  and  products  side  by 
side  and  compare  them. 


54  MULTIPLICATION. 

3.  How  may  the  product  be  found  from  the  multiplicand 
when  the  multiplier  is  10  ? 

4.  How   many  are    100   times   2?     3?     4?     5?     6?     7? 
8?     9? 

5.  How  may  any  number  be  multiplied  by  100?  by  1000? 

6.  How  may  a  number  be  multiplied  by  1  with  any  num- 
ber of  ciphers  affixed  ? 

80.  PRINCIPLE. — In  multiplying  by  10,  100,  1000,  etc.,  as 
many  ciphers  must  be  annexed  to  the  right  of  the  multiplicand  as 
there  are  ciphers  in  the  multiplier. 

1.  Multiply  36  by  1000. 

PROCESS.  ANALYSIS. — Since  in  multiplying  by  1  with    any 

3  6  number   of    ciphers    annexed,    we    annex    as    many 

1000  ciphers  to  the  multiplicand  as  there  are  in  the  mul- 
tiplier, to  multiply  by  1000  we  annex  three  ciphers 
to  the  multiplicand,  which  gives  the  product  36000. 

2.  Multiply  2360  by  400. 

PROCESS.  ANALYSIS. — Since  2360  is  equal  to  236  X  10>  and 

2360  40°  is  e<lual  to  4  X  100> the  Product  of  236°  X  400  may 

J.  0  0        ^e  obtained  by  multiplying  236  by  4,  and  this  product 

i-LjL      by  10  times  100,  or  1000.     The  product  of  236  X  4  is 

944000      944^  and  this  may  be  multiplied  by  1000  by  annexing 
three  ciphers  (Prin.),  giving  as  a  result  944000. 

RULE. — Multiply  without  regard  to  the  ciphers  on  the  right, 
and  to  the  product  annex  as  many  ciphers  as  there  are  on  the 
right  of  both  multiplier  and  multiplicand. 

3.  Multiply    375  by    10.  By    100.  By      40.  By    300. 

4.  Multiply    845  by    30.  By      70.  By    600.  By    900. 

5.  Multiply    176  by  500.  By    700.  By    400.  By  1000. 

6.  Multiply  1385  by  200.  By  2000.  By  2200.  By  3300. 

7.  Multiply  4860  by  250.  By  3200.  By  4200.  By  6500. 
8    Multiply  3120  by  210.  By  3800.  By  2700.  By  4600. 


MULTIPLICATION.  55 

9.  In  a  mile  there  are  5280  feet.     How  many  feet  are 
there  in  500  miles? 

10.  In  an  acre   there   are  160  square  rods.     How  many 
square  rods  are  there  in  a  farm  of  300  acres? 

11.  A  farmer  sold  a  flock  of  260  sheep  at  $3.20  per  head. 
How  much  did  he  get  for  them  ? 

12.  A  drover  sold  1120  hogs  at  an  average  price  of  816.30 
per  head.     How  much  did  he  receive  for  them? 

EXAMPLES. 

81.    1.  What  will  be  the  cost  of  896  chests  of  tea,  each 
chest  containing  58  pounds,  at  63  cents  a  pound? 

2.  An  agent  sold  3923  Lyman's  Historical  Charts  at  $3.50 
each.     How  much  did  he  receive  for  them? 

3.  I  have  6  bins  that  hold  119  bushels  each.     They  are 
full  of  grain  and  I  have  already  sold  515  bushels.     It  was  all 
raised  on  my  farm  this  year.     How  much  grain  was  raised? 

4.  A.  J.  Newton  &  Co.  bought  113  cases  of  calico,  each 
case  containing  64  pieces,  and  each  piece  47  yards.     How 
many  yards  did  they  buy? 

5.  A  drover  bought  25  oxen  at  $85  a  head,  316  sheep  at 
$4.50  a  head,  and  94  calves  at  $8  a  head.     What  was  the 
whole  amount  paid? 

6.  A  man  insured  2  houses  valued  at  $3750  and  $4650, 
respectively,  at  the  rate  of  $2  per  hundred  dollars.     How 
much  did  the  insurance  cost  him? 

7.  If  I  have  219  acres  of  land,  and  each  acre  produces 
47  bushels  of  corn,  how  many  bushels  do  I  receive? 

8.  How  many  quills  can  be  obtained  from  398  geese,  if 
each  wing  furnishes  6  quills? 

9.  A  grocer  sold  in  one  month  81  dozen  eggs  at  26  cents 
per  dozen ;  in  the  next,  53  dozen  at  28  cents  per  dozen.    How 
much  money  did  he  receive  for  the  eggs? 


56  MULTIPLICATION. 

10.  It  requires  1716  pickets  to  fence  one  side  of  a  square 
lot.     How  many  pickets  will  be  required  to  fence  13  lots  of 
the  same  size  and  shape? 

11.  A  sold  13  firkins  of  butter,  each  firkin  containing  56 
pounds,  at  $  .34  a  pound.     How  much  did  he  receive  for  it? 

12.  A  coal  dealer  bought  13  car  loads  of  coal,  each  load 
containing  10  tons,  at  $6.85  a  ton.     He  retailed  48  tons  of 
this  at  $7  per  ton,  28  tons  at  $8.25  per  ton,  27  tons  at  $8.75 
per  ton,  and  the  remainder  at  $9.50  per  ton.     How  much  did 
he  make  by  the  transaction? 

13.  An  army  lost  in  battle  315  killed,  417  wounded;  the 
enemy  lost  in  killed  and  wounded,  together,  13  times  as  many. 
How  many  soldiers  were  killed  and  wounded  in  this  battle? 

14.  If  two  steamers  should  leave  New  York  at  the  same 
time,  and  should  sail  in  the  same  direction,  the  first  at  the 
rate  of  18  miles  an  hour,  the  second  at  the  rate  of  15  miles 
an  hour,  how  far  apart  would  they  be  in  36  hours? 

15.  Mr.  Hudson  bought  350  bushels  of  corn  at  65  cents  a 
bushel,  215  bushels  of  wheat  at  $1.35  per  bushel,  and  273 
bushels  of  oats  at  43  cents  a  bushel.     What  did  the  whole 
cost  him? 

16.  Mr.  Henderson  sold  a  farm  of  325  acres  at  $65.50  per 
acre,  and  received  in  payment  345  sheep  at  $3.25  per  head,  a 
note  for  $2684.95  and  the  rest  in  cash.     How  much  cash  did 
he  receive? 

17.  A  cloth  merchant  sold  two  lots  of  cassimeres,  the  first 
containing  17  pieces  of  28  yards  each,  at  $1.75  per  yard,  the 
second  containing  23  pieces  averaging  29  yards  each,  at  $1.85 
per  yard.     What  was  the  value  of  the  whole  ? 

18.  An  excursion  train  composed  of  13  passenger  coaches, 
each  containing  37  persons,  went  from  Syracuse  to  Niagara 
Falls  and  back.     If  the  fare  to  Niagara  Falls  and  return  to 
Syracuse,  was  $3.25  per  ticket,  how  much  did  the  railroad 
company  receive? 


I  S  I  O  tsl 


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C-T— >  =;~~  ~"^ 

INDUCTIVE    EXERCISES. 

82.    1.  How  many  groups  of  2  birds  each  can  be  formed 
from  6  birds?     How  many  2's  are  there  in  6? 

2.  How  many  groups   of  3   sheep  each  can   be  formed 
from  9  sheep?     How  many  3's  are  there  in  9? 

3.  How  many  groups  of  2  chickens  each  can  be  formed 
from  10  chickens?     How  many  2's  are  there  in  10? 

4.  At  5  cents  apiece,  how  many  pencils  can  be  bought 
for  10  cents?     How  many  5's  are  there  in  10? 

5.  When  milk  is  worth  7  cents  a  quart,  how  many  quarts 
can  be  bought  for  28  cents?     How  many  7's  are  there  in  28? 

6.  There  are  20  panes  of  glass  in  the  front  of  a  block  of 
stores.     If  each  window  contains  4  panes,  how  many  win- 
dows are  there?     How  many  4's  are  there  in  20? 

7.  At  8  cents  a  dozen,  how  many  peaches  can  be  bought 
for  24  cents?     How  many  times  8  cents  are  24  cents?     How 
many  8's  are  there  in  24? 

8.  How  many  groups  of  4  things  each  can  be  formed 
from  16  things?     How  many  4's  are  there  in  16? 

9.  A  merchant  had  30  yards  of  calico  which  he  cut  into 
pieces  5  yards  long.     How  many  pieces  did  it  make?     How 
many  5's  are  there  in  30?     How  many  times  is  5  contained 
in  30? 

10.  How  many  9's  are  there  in  18?     How  many  times  is 
9  contained  in  18? 

(57) 


58  DIVISION. 

11.  How  many  5's  are  there  in  10?     In  15?     In  20?     In 
25?     In  30? 

12.  If  15  cents  are  divided  equally  among  3  boys,  how 
many  cents  will  each  receive?     When  15  cents  are  divided 
into  3  equal  parts,  how  many  cents  will  each  part  contain  ? 

13.  If  12  peaches  are  arranged  in  3  rows,  how  many  will 
there  be  in  each  row? 

14.  What  is  one  of  the  4  equal  parts  of  8?    Of  12?    Of  16? 

15.  How  many  3's  are  there  in  30?     How  many  are  10 
threes,  or  10  times  3? 

16.  How  many  4's  are  there  in  40?     How  many  times  is 
4  contained  in  40?     How  many  are  10  fours? 

DEFINITIONS. 

83.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another;  or, 

The  process  of  separating  a  number  into  equal  parts. 

84k  The  Dividend  is  the  number  to  be  divided. 

85.  The   Divisor   is  the  number  by  which  we  divide. 
It  shows  into  how  many  equal  parts  the  dividend  is  to  be 
divided. 

86.  The  Quotient  is  the  result  obtained  by  division. 
It  shows  how  many  times  the  divisor  is  contained  in   the 
dividend. 

87.  The  part  of  the  dividend  remaining  when  the  division 
is  not  exact  is  called  the  Remainder. 

88.  The  Sign  of  Division  is  -r- .     It  is  read  divided 
by.     When  placed  between  two  numbers  it  shows  that  the 
one  at  the  left  is  to  be  divided  by  the  one  at  the  right. 

Thus,  154  -r-  7,  is  read  154  divided  by  7. 


DIVISION.  59 

Division  is  also  indicated  by  placing  the  dividend  above 
the  divisor  with  a  line  between  them,  and  by  writing  the 
divisor  at  the  left  of  the  dividend  with  a  curved  line  between 
them.  Thus,  154  divided  by  7,  may  also  be  written  -!-p,  and 
7)154. 

89.  PRINCIPLES. — 1.  The  dividend  and  divisor  must  be  like 
numbers. 

2.  The  quotient  must  be  an  abstract  number. 

3.  The  product  of  the  divisor  by  the  quotient,  plus  the  remain- 
der, is  equal  to  the  dividend. 

9  1.  In  problems  where  it  is  required  to  separate  a  number 

3        into  equal  parts,  it  is  customary  to  regard  the  dividend  and 
—      quotient    as   like   numbers,    and   the   divisor   as   an   abstract 
number. 

2.  The  example  "How  many  3's  are  there  in  9"  may  be 
3  solved,  as  in  the  margin,  by  subtraction.  All  examples  in  divis- 
3  ion  may  be  solved  in  the  same  manner.  Hence,  division  may 

be  regarded  as  a  shwt  method  of  subtracting  equal  numbers. 
3.  In  multiplication  two  numbers  are  given  to  find  their  product, 
In  division  the  product  is  given  and  one  of  the  factors  to  find  the  other. 
Hence,  division  is  the  converse  of  multiplication. 


CASE  I. 
90,  When  the  divisor  is  expressed  hy  one  figure. 

1.  At  $8  each, how  many  plows  can  be  bought  for  $24? 

ANALYSIS. — Since  each  plow  costs  $8,  as  many  plows  can  be  bought 
for  $24  as  $8  is  contained  times  in  $24,  which  is  3  times.  Therefore 
3  plows  can  be  bought  for  $24. 

2.  If  a  man  can  earn  $7  in  a  day,  how  long  will  it  take 
him  to  earn  $28? 

3.  At  $4  each,  how  many  hats  can  be  bought  for  $24? 

4.  When  flour  is  selling  at  $6  a  hundred-weight,  how  many 
hundred-weight  can  be  bought  for  $36? 


60  DIVISION. 

5.  If  a  mason  built  3  rods  of  walk  per  day,  how  long  did 
it  take  him  to  build  21  rods? 

6.  B  paid   96  cents  for  glass  at  8  cents  a  pane.     How 
many  panes  did  he  buy? 

7.  At  $9  a  cord,  how  many  cords  of  wood  can  be  bought 
for  $45? 

8.  If  a  man  earns  $11  a  week,  how  many  weeks  will  he 
require  to  earn  $66? 

9.  How  many  lots  of  11  acres  each  can  be  made  from  a 
farm  containing  132  acres? 

10.  If  a  farmer  exchanges  6  firkins  of  butter  worth  $20  a 
firkin  for  cloth  at  $4  a  yard,  how  many  yards  will  he  receive? 

11.  My  coal  cost  me  $35  at  the  rate  of  $7  a  ton.     How 
many  tons  did  I  purchase? 

12.  How  many  engravings  must  an  artist  sell  for  $12  apiece 
to  realize  $84? 

13.  When   sugar  is  worth   9  cents  a  pound,  how  many 
pounds  can  be  bought  for  45  cents? 

14.  At  the  rate  of  $7  a  rod,  how  many  rods  of  fence  can 
be  built  for  $63? 

15.  I  hired  a  man  for  $45  to  do  a  piece  of  work  at  the  rate 
of  $5  a  day.     How  many  days  did  it  take  him  ? 

16.  A  lady  bought  some  silk  worth  $3  a  yard,  paying  $36 
for  it.     How  many  yards  did  she  buy? 

17.  How  many  barrels  of  flour  at   $8  a  barrel  can  be 
bought  for  $48? 

18.  How  many  pounds  of  nails  can  be  bought  for  75  cents 
at  the  rate  of  4  pounds  for  20  cents? 

19.  I  bought  6  sheep  for  $30.     How  much  did  I  pay  per 
head? 

20.  At  $5  per  head,  how  many  head  of  sheep  can  be  bought 
for  $37?  Am.  7  sheep  and  $2  left. 

21.  A  man  whose  wages  were  $4  a  day  earned  in  a  certain 
time  $33.    How  many  days  did  he  work?        Am.  8^  days. 


DIVISION.  61 

91.  From  examples  20   and   21   it  is   apparent   that   the 
remainder  may  be  written  either  after  the  quotient,  as  in  the 
answer  to  the  20th,  or  as  a  part  of  it,  as  in  the  answer  to 
the  21st, 

When  written  as  a  part  of  the  quotient,  the  remainder  is 
expressed  by  placing  the  divisor  under  it  with  a  line  between 
them.  Such  an  expression  shows  that  each  unit  of  the  re- 
mainder is  to  be  divided  into  as  many  equal  parts  as  there 
are  units  in  the  divisor. 

When  any  thing  is  divided  into  two  equal  parts,  each  of  the 
parts  is  called  one  half. 

When  into  three  equal  parts,  each  part  is  called  one  third. 

When  into  four  equal  parts,  each  part  is  called  one  fourth. 

When  into  five,  six,  seven,  etc.,  equal  parts,  the  parts  are 
called  fifths,  sixths,  sevenths,  etc. 

^  expresses  one  half,  or  one  of  two  equal  parts  of  any 
thing. 

^  expresses  one  fourth,  or  one  of  four  equal  parts  of  any 
thing. 

f  expresses  two  fifths,  or  two  of  five  equal  parts  of  any 
thing. 

2^  expresses  five  twenty-sevenths,  or  five  of  twenty-seven 
equal  parts  of  any  thing. 

92.  One  or  more  of  the  equal  parts  of  any  thing  is  called 
a  Fraction. 

93.  Read  the  following  fractional  expressions: 


A          ft         1*         It 

VV  W  A  ft 


T 


22.  If  James  should  divide  25  apples  equally  among  5 
boys,  what  part  of  the  whole  would  each  receive?  How 
many  apples  would  each  receive? 


62 


DIVISION. 


ANALYSIS. — If  he  should  divide  25  apples  equally  among  5  boys, 
each  boy  would  receive  (me-fifth  of  25  apples,  which  is  5  apples. 

23.  If  flour  is  worth  $8  a  barrel,  what  will  one-half  barrel 
cost? 

24.  Mr    Smith   bought   8  bushels   of  chestnuts  for   $24. 
How  much  did  he  pay  per  bushel?     How  much  is  one-eighth 
of  24? 

25.  What  is  one-sixth      of  36?     Of  42? 

26.  What  is  one-tenth      of  50?     Of  60? 

27.  What  is  one-seventh  of  14?     Of  28? 


Of  48?  Of  60? 
Of  70?  Of  80? 
Of  42?  Of  49? 


WRITTEN    EXERCISES. 


94.    1.  Divide  1396  by 

1ST  PROCESS. 
Divisor.  Dividend.  Quotient. 

4)1396(300  times. 
1200       40  times. 


196 

160 

36 

36 


9  times. 


349  times. 


2D  PROCESS.  1  1  1 

4)1396(349 
12 


19 
16 


36 


the  partial  dividend,  there  is 


4. 

ANALYSIS. — For  convenience  we 
write  the  divisor  at  the  left,  and  the 
quotient  at  the  right  of  the  dividend, 
with  curved  lines  between  them,  and 
begin  at  the  left  to  divide. 

4  is  not  contained  in  1  thousand 
any  thousand  times,  therefore  the 
quotient  can  not  contain  units  of  any 
order  higher  than  hundreds.  Hence 
we  find  how  many  times  4  is  con- 
tained in  all  the  hundreds  of  the 
dividend.  1  thousand  plus  3  hun- 
dreds equals  13  hundreds.  4  is  con- 
tained in  13  hundreds  3  hundred 
times  and  a  remainder.  We  write 
the  3  hundreds  in  the  quotient  and 
multiply  the  divisor  by  it,  obtaining 
for  a  product  12  hundreds,  or  1  thou- 
sand 2  hundred,  which  we  write  under 
units  of  the  same  order  in  the  divi- 
dend. Subtracting  this  product  from 
a  remainder  of  1  hundred. 


DIVISION.  63 

1  hundred  plus  9  tens  equals  19  tens.  4  is  contained  in  19  tens  4 
tens  times  and  a  remainder.  We  write  the  4  tens  in  the  quotient  and 
multiply  the  divisor  by  it,  obtaining  for  a  product  16  tens,  or  1  hundred 
and  6  tens,  which  we  write  under  units  of  the  same  order  in  the  partial 
dividend.  Subtracting,  there  is  a  remainder  of  3  tens  and  6  units. 

3  tens  plus  6  units  equals  36  units.  4  is  contained  in  36  units  9 
times.  We  write  the  9  units  in  the  quotient  and  multiply  the  divisor 
by  it,  obtaining  for  a  product  36  units,  or  3  tens  and  6  units,  which  we 
write  under  units  of  the  same  order  in  the  partial  dividend.  Subtract- 
ing, there  is  no  remainder.  Hence  the  quotient  is  349. 

In  the  second  process  all  ciphers  are  omitted  from  the  right  of  the 
products  and  the  significant  figures  are  written  under  units  of  the  same 
order.  The  quotient  also  is  expressed  by  writing  the  different  orders  of 
units  in  proper  succession. 

PROOF. — 349  the  quotient,  multiplied  by  4  the  divisor,  is  equal  to 
1396  the  dividend. 

Hence  the  work  is  correct.     (Prin.  3.) 


Solve  in  like  manner  and  prove: 


2.  738  -r-  3. 

3.  845 -- 5. 

4.  385-*- 7. 


5.  4821  —  3. 

6.  3462  —  6. 

7.  3864  —  8. 


8.  7848  -f-9. 

9.  8432 --4. 
10.  8308  -r-7. 


95.  The  solution  of  the  preceding  examples  may  be  short- 
ened by  performing  the  multiplications  and  subtractions  with- 
out writing  the  results.  This  process  is  called  Short 
Division. 

The  solution  of  Example  1  by  Short  Division  is  as  follows: 

PROCESS.   .  ANALYSIS. — 4  is  contained  in  13  hundred  3 

4)1396  hundred  times  and  1  hundred  remainder.     We 

— — —  write  3  hundreds  in  the  quotient  under  units 

of  the  same  order  in  the  dividend.     1  hundred 

remainder  united  with  9  tens  makes  19  tens. 

4  is  contained  in  19  tens  4  tens  times  and  3  tens  remainder.  We  write 
the  4  tens  in  the  quotient  under  tens  of  the  dividend.  3  tens  remain- 
der united  with  6  units  make  36  units.  4  is  contained  in  36  units  9 
times.  We  write  the  9  in  the  quotient.  Hence  the  quotient  is  349. 


64 


DIVISION. 


Solve  by  short  division  .• 

11.  4872- 

-4. 

17. 

12.  6830- 

-5. 

18. 

13.  2976- 

-6. 

19. 

14.  2985- 

-5. 

20. 

15.  4635- 

-3. 

21. 

16.  3936- 

-4. 

22. 

23.  4567- 

-5. 

24.  8932- 

-6. 

25.  8174- 

-9. 

26.  9185- 

-4. 

27.  8436- 

-7. 

28.  3885- 

-8. 

CASE  II. 

96.  When  the  divisor  is  expressed  by  more  than 
one  figure. 

1.  How  many  barrels  of  flour  at  $10  a  barrel  can  be  bought 

for  $80? 

ANALYSIS. — Since  1  barrel  costs  $10,  as  many  barrels  can  be  bought 
for  $80  as  $10  are  contained  times  in  $80  which  is  8  times.  Therefore 
8  barrels  can  be  bought  for  $80. 

2.  How  many  pounds  of  mutton  at  10  cents  a  pound  can 
be  bought  for  50  cents?     How  many  10's  in  50?     In  60^ 
In  70?     In  80? 

3.  A  man  measured  a  stick  and  found  it  to  be  60  inches 
long.     There  are  12  inches  in  a  foot.     How  many  feet  long 
was  it?     How  many  12's  in  60?     In  72?     In  84?     In  96? 

4.  At  $13  a  ton  how  much  hay  can  be  bought  for  $26? 

5.  At  15  cents  each  how  many  toys  can  be  bought  for  30 
cents?     For  45  cents?     For  60  cents? 

6.  Mr.  Henderson  sold  20  lambs  for  $80.     How  much 
did  he  get  apiece  for  them? 

7.  25  cents  make  a  quarter  of  a  dollar.      How   many 
quarters  of  a  dollar  has  a  boy  who  has  50  cents? 

8.  Henry's  father  gave  him  a  dollar.     How  many  pine- 
apples at  20  cents  each  can  he  buy  with  the  money? 

9.  The  railroad  fare  to  a  certain  place  is  35  cents.     How 
many  tickets  can  be  bought  with  70  cents? 


DIVISION.  65 

10.  If  a  boy  earns  11  cents  an  hour,  how  long  will  it  take 
him  to  earn  55  cents?     66  cents?     88  cents? 

11.  There  are  20  hundred-weight  in  a  ton.     How  many 
tons  are  there  in  45  hundred-weight?     How  many  in  55? 

12.  In  one  day  there  are  24  hours.     How  many  days  are 
there  in  50  hours?     In  60  hours?     In  72  hours? 

13.  12  articles  make  a  dozen.     How  many  dozen  are  there 
in  39  articles?     In  48?     In  51?     In  60?  '  In  65? 

14.  A  farm  of  60  acres  was  divided  into  15  equal  lots. 
How  many  acres  were  there  in  each  lot? 

15.  At  18  cents  a  dozen,  how  many  dozen  of  eggs  can  be 
bought  for  36  cents?     For  40  cents?     For  54  cents? 

16.  When  butter  is  30  cents  a  pound,  how  many  pounds 
can  be  bought  for  90  cents?     For  $1:20?     For  $1.50? 

17.  How  many  20's  are  there  in  40?     In  50?     In  60? 

18.  How  many  25's  are  there  in  50?     In  60?     In  75? 

19.  By  selling  brooms  at  25  cents  each,  I  received  $1.25. 
How  many  brooms  did  I  sell? 

20.  In  100  how  many  10's  are  there?     How  many  ITs? 
12's?     13's?     15's?     16's?    20's?    23's?    25's? 

WRITTEN    EXERCISES. 

97.    1.  Divide  7975  by  26. 

PROCESS.  ANALYSIS. — 26  is  not  contained  in  7 

Divisor.  Dividend.  Quotient.          thousands  any  thousands    times;   hence 
26)7975(306-1^-        we  unite  the  thousands  with  the  hun- 
rj  g  dreds,  making  79  hundreds.     26  is  con- 

tained in  79  hundreds  3  hundred   times 
1  •  5  an(j  a  remainder.     We  write  the  3  hun- 

156  dreds  in  the  quotient  and  multiply  the 

^  9  divisor  by  it,  obtaining  for  a  product  78 

hundreds,  or  7  thousands    and    8   hun- 
dreds, which  we  write  under  units  of  the  same  order  in  the  dividend. 
Subtracting,  there  is  a  remainder  of  1  hundred. 
5 


66  DIVISION. 

We  unite  the  1  hundred  with  the  7  tens,  making  17  tens.  26  is 
not  contained  in  17  tens  any  tens  times ;  therefore  there  are  no  tens 
in  the  quotient,  and  we  write  a  cipher  there. 

We  unite  the  17  tens  with  the  5  units,  making  175  units.  26  is 
contained  in  175  units  6  times  and  a  remainder.  We  write  the  6  in 
units7  place  in  the  quotient  and  multiply  the  divisor  by  it,  obtain- 
ing for  a  product  156  units,  or  1  hundred,  5  tens,  and  6  units,  which 
we  write  under  units  of  same  order  in  the  partial  dividend.  Sub- 
tracting, there  is  a  remainder  of  19.  We  write  the  remainder  over 
the  divisor  as  a  part  of  the  quotient. 

Hence  the  quotient  is  306  |f. 

PROOF.— 306X26  +  19  =  7975.  Hence  the  work  is  correct.  (Prin.  3.) 

98.  When  the  steps  in  the  solution  of  an  example  in  divis- 
ion are  written,  the  process  is  called  Long  Division. 

RULE. —  Write  the  divisor  at  the  left  of  the  dividend  with  a 
curved  line  between  them. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest 
figures  on  the  left  hand  of  the  dividend  that  will  contain  it,  and 
write  the  quotient  on  the  right. 

Multiply  the  divisor  by  this  quotient  and  place  the  product  under 
the  figures  divided.  Subtract  the  result  from  the  partial  dividend 
used,  and  to  the  remainder  annex  the  next  figure  of  the  dividend. 

Divide  as  before  until  all  the  figures  of  the  dividend  have  been 
annexed  to  the  remainder. 

If  any  partial  dividend  will  not  contain  the  divisor,  write  a 
cipher  in  the  quotient,  then  annex  the  next  figure  of  the  dividend 
and  proceed  as  before. 

If  there  is  a  remainder  after  the  last  division  write  it  after  the 
quotient,  or  with  the  divisor  under  it  as  part  of  the  quotient. 

PROOF. — Multiply  the  divisor  by  the  quotient,  and  to  the  prod- 
uct add  the  remainder,  if  any.  If  the  work  is  correct,  the  result , 
ivill  equal  the  dividend. 

1.  To  find  the  quotient  figure,  see  how  many  times  the  first  figure  of  the 
divisor  is  contained  in  first  figure*  of  the  partial  dividend  that  will  con- 


DIVISION. 


67 


tain  it,  making  allowance  for  the  addition  of  the  tens  from  the  prod- 
uct of  the  second  figure  of  the  divisor. 

2.  If  the  product  of  the  divisor  by  the  quotient  figure  be  greater 
than  the  partial  dividend  from  which  it  is  to  be  subtracted,  the  quo- 
tient figure  is  too  large. 

3.  Each  remainder  must  be  less  than  the  divisor;   otherwise  the 
quotient  figure  is  too  small. 

4.  When  there  is  no  remainder  the  divisor  is  said  to  be  exact. 


EXAMPLES. 

99.  Divide: 

Divide  : 

2.  1240  by  10. 

25. 

12456  by  24. 

3.  3443  by  11. 

26. 

28350  by  54. 

4.  2592  by  12. 

27. 

50854  by  94. 

5.  3978  by  13. 

28. 

58176  by  96. 

6.  5684  by  14. 

29. 

56394  by  78. 

7.  6480  by  15. 

30. 

54944  by  101. 

8.  8736  by  21. 

31. 

90992  by  121. 

9.  1472  by  32. 

32. 

199864  by  301. 

10.  9672  by  31. 

33. 

475524  by  612. 

11.  9724  by  22. 

34. 

1445204  by  802. 

12.  2952  by  72. 

35. 

1760225  by  905. 

13.  1188  by  54. 

36. 

3156584  by  722. 

14.  4235  by  55. 

37. 

5173302  by  834. 

15.  5356  by  52. 

38. 

5926431  by  643. 

16.  8733  by  41. 

39. 

3214664  by  566. 

17.  9639  by  81. 

40. 

6923471  by  555. 

18.  7991  by  61. 

41. 

14293624  by  675. 

19.  2508  by  22. 

42. 

56243121  by  686. 

20.  7332  by  52. 

43. 

692348726  by  897. 

21.  4824  by  72. 

44. 

496839715  by  1047. 

22.  16665  by  33. 

45. 

786935846  by  3118. 

23.  13545  by  43. 

46. 

1234589640  by  96813. 

24.  25578  by  63. 

47. 

31964875932  by  37425. 

68  DIVISION. 

48.  Into  how  many  lots  of  39  acres  each,  can  a  tract  of 
land  containing  6318  acres  be  divided? 

49.  Wm.  Wallace  has  17  horses,  the  aggregate  value  of 
which  is  $4386.     What  is  the  average  worth  of  each  horse? 

50.  A  surveyor  traveled  41600  rods  in  one  week.     How 
many  miles  did  he  travel,  there  being  320  rods  in  a  mile? 

51.  How  many  eggs  at  $.38  per  dozen  can  be  bought  for 
$6.84? 

52.  In  24  hours  the  earth  moves  1575000  miles.     How  far 
does  it  move  in  one  minute,  60  minutes  making  an  hour? 

53.  Mount  Everest  in  Asia  is  29100  feet  high.     There  are 
5280  feet  in  a  mile.     How  many  miles  high  is  it? 

54.  It  required  4375480  bricks  to  build  an  orphan  asylum. 
How  many  days  did  it  require  5  teams  to  draw  the  bricks,  if 
they  drew  5  loads  per  day  and  1250  bricks  at  a  load? 

55.  The  earth  is  91500000  miles  from  the  sun.    How  many 
seconds  does  it  take  light  to  come  from  the  sun  to  the  earth, 
if  it  travels  185000  miles  per  second? 

56.  A  man  bought  a  farm  of  278  acres  at  $63  an  acre. 
He   paid   $1275.   down   and   agreed    to    pay   the   rest  in    8 
equal  annual  payments.     How  much  was  he  required  to  pay 
yearly? 

57.  The  earnings  of  a  certain  railroad  were  $3681452  during 
the  year.     The  number  of  days  in  a  year  is  365.     What  was 
the  average  income  per  day? 

58.  How  many  feet  are  there  in  a  mile,  if  42  miles  contain 
221760  feet? 

59.  If  the  average  wages  of  a  laboring  man  are  $500  per 
year,  how  many  men  will  it  require  to  earn  $50000  per  year, 
Jhe  salary  of  the  President? 

60.  The  area  of  the  State  of  North  Carolina  is  50704  square 
miles,  and  the  population,  according  to  the  census  of  1870, 
was  1071400.     How  many  persons,  on  an  average,  were  there 
living  on  a  square  mile? 


DIVISION.  69 

CASE  III. 

100.  When  the  divisor  has  ciphers  on  the  right. 

1.  How  many  10's  and  what  remainder  in  46?     84?     97? 

2.  When  a  number  is  divided   by  10,  what  part  is  re- 
mainder?    What  part  is  quotient? 

3.  How  many  hundreds,  and  what  remainder  in  434?     516? 
639?     758? 

4.  When  a  number  is  divided  by  100  what  part  of  it  is 
remainder?     What  part  is  quotient? 

*5.  When  a  number  is  divided  by  1000,  what  part  is  re- 
mainder?    What  part  is  quotient? 

101.  PRINCIPLE.—  In  dividing  by  10,  100,  1000,   etc.,  the 
remainder  will  be  as  many  of  the  figures  at  the  right  of  the  divi- 
dend as  there  are  ciphers  on  the  right  of  the  divisor.     The  rest  of 
the  number  is  quotient. 

102.  1.  Divide  6374  by  1000. 

PROCESS.  ANALYSIS.  —  Since  the  divisor  contains  no 

11000^61374         order  of  units  lower  than  thousands,  in  divid- 

ing we  may  omit  or  cut  off  from  the  dividend 

6  iVcrb"         for  a  remainder,  all  orders  of   units  lower 

than  thousands.     (Prin.) 

Dividing  6  thousands  by  1  thousand  we  obtain  6  for  the  quotient 
and  374  for  the  remainder  or 


2.  Divide  39321  by  6000. 

ANALYSIS.  —  Since  the  divisor  con- 

i  tains  no  order  of  units  lower  than 

i  thousands,  in  dividing  we  may  omit 

6      Rem.  3000      or  cut  off  from  the  dividend  all  or- 
321      ders   of   units  lower  than  thousands. 

6  thousands   are  contained  in  39 

Entire  remainder  6621,,  ,      „    ^.  jo^  j 

thousands   6   times   and   3    thousand 

remainder.     3  thousand  plus  the  other  partial  remainder  equals  the 
entire  remainder.     Hence  the  quotient  is 


70  DIVISION. 

RULE. — Cut  off  the  ciphers  from  the  right  of  the  divisor,  and  as 
many  figures  from  the  right  of  the  dividend. 

Divide  the  rest  of  the  dividend  by  the  rest  of  the  divisor. 

Annex  to  the  remainder  the  figures  cut  off:  the  result  will  be 
the  true  remainder. 


Divide  the  following: 

3.  Divide    1869  by  100. 

4.  Divide  12345  by  200. 

5.  Divide  89325  by  700. 

6.  Divide  35968  by  900. 


7.  Divide    2465  by  1000. 

8.  Divide  13692  by  4000. 

9.  Divide  83005  by  1100. 
10.  Divide  75684  by  1500. 


11.  How  many  miles  of  railroad  at  $50000  a  mile  can  be 
constructed  for  $38968457? 

12.  How  many  schooners  carrying  8300  bushels  of  wheat 
will  it  require  to  carry  984364  bushels? 

13.  The  area  of  the  State  of  New  York  is  47000  square 
miles,  or  30080000  acres.     How  many  acres  in  a  square  mile? 

103.  RELATION  OF  DIVIDEND,  DIVISOR,  AND  QUOTIENT. 

The  value  of  the  quotient  depends  upon  that  of  the  divi- 
dend and  divisor.  If  one  of  these  is  changed,  while  the  other 
remains  the  same,  the  quotient  will  be  changed.  If  both  are 
changed,  the  quotient  may  not  be  changed. 

The  changes  may  be  illustrated  as  follows : 

FUNDAMENTAL   EQUATION. 

64  -=-8  =  8. 


CHANGED   EQUATIONS. 
1.  128  -5-  8  =  16  "I     1.  Multiplying  the  dividend 


changed. 


1     D*  V7     rl  '  by  2   multiplies  the  quotient 

\  by  2. 

2.     32  -H  8  =    4  |      2.  Dividing  the  dividend  by 
2  divides  the  quotient  by  2. 


DIVISION. 


71 


2.  Divisor 
changed. 


3.  Both 
changed. 


1.  64 --16:=   4 

2.  64  -*-    4  =  16 

1.  128-*- 16  =  8 

2.  32--   4  =  8 


1.  Multiplying  the  divisor 
by  2  divides  the  quotient  by 
2. 

2.  Dividing  the  divisor  by 
2  multiplies  the  quotient  by  2. 

Multiplying  or  dividing 
both  dividend  and  divisor  by 
2  does  not  change  the  quo- 
tient. 


From  these  illustrations  the  following  principles  are  de- 
duced : 

104.  PRINCIPLES. — 1.  Multiplying  the  dividend  or  dividing 
the  divisor,  multiplies  the  quotient. 

2.  Dividing  the  dividend  or  multiplying  the  divisor,  divides  tlie 
quotient. 

3.  Multiplying  or  dividing  both  dividend  and  divisor  by  the 
same  number,  does  not  change  the  quotient. 


ANALYSIS    AND   KEVIEW. 

105.  Analysis  is  the  process  of  solving  problems  by 
tracing  the  relation  of  the  parts. 

In  analyzing  we  commonly  reason  from  the  given  number  to 
one,  and  then  from  one  to  the  required  number. 

1.  If  8  yards  of  cloth  cost  $16,  what  will  12  yards  cost? 

PROCESS.  ANALYSIS. — Since  8  yards  cost  $16>  1  yard 

8  yards=$16.  will  cost  one-eighth  of  $16,  or  $2;   and  since 

1       "    — $  2.  1  yard  costs  $2,  12  yards  will  cost  12  times 

12       "    =$24.  $2,  or  $24. 

2.  If  8  horses  cost  $2400,  what  will  6  horses  cost? 

3.  If  8  lemons  cost  40  cents,  what  will  11  lemons  cost? 

4.  How  much  will  12  hats  cost,  if  8  hats  cost  $16? 


72  DIVISION. 

5.  If  25  pounds  of  sugar  cost  $2.50,  what  will  36  pounds 
cost? 

6.  If  12  men  can  build  a  school-house  in  25  days,  how  long 
will  it  take  25  men  to  build  it  ? 

7.  If  12  barrels  of  flour  are  worth  $132,  what  are  22  bar- 
rels worth? 

8.  If  it  requires  576  feet  of  boards  to  build  18  rods  of 
fence,  how  many  feet  will  be  required  to  build  13  rods? 

9.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  it? 

10.  If  I  exchanged  18  barrels  of  flour  for  61  yards  of  cloth 
at  $4  a  yard,  how  much  did  I  get  per  barrel  for  the  flour? 

106.  The  Parenthesis,  (  ),  shows  that  the  numbers 
included  within  it  are  to  be  subjected  to  the  same  operation. 

Thus,  ( 5  +  6  —  2 )  X  3  shows  that  5  +  6  —  2,  or  9,  is  to  be  multi- 
plied by  3. 

107.  The  Vinculum,  "     ~~,  may  be  used  instead  of 
the  parenthesis. 

Thus,  instead  of  ( 5  +  6  —  2 )  X  3,  we  may  write  5  +  6— 2X 3. 

Find  the  value  of  the  following : 

11.  (12  +  7  — 9)  X  5. 

12.  (13  — 6  +  8)x6. 

13.  (11  —  2  +  5)  X8. 

14.  (3  +  4)x9  — (3  +  6)-3. 

15.  (5  +  7  — 3)x3  +  (3  +  5  — 4)-v-4. 


16.  (36  —  7)x5  +  (102  +  6)~ 9. 


17.  (  99  —  3  )  -i-  8  —  (  86  +  10  )  —  12  +  (  3  +  6  )  -^  3. 
18. 


-(7  — 2)  +  6. 

19.  A  man  dying,  left  the  following  tracts  of  land  to  be 
divided  equally  among  his  five  children.  The  first  tract  con- 
tained 1118  acres;  the  second,  3  times  as  much  lacking  193 


DIVISION.  73 

acres;  the  third,  twice  as  much  as  the  other  two  lacking  105 
acres.     What  was  each  one's  share? 

20.  A  gentleman  bought  1516  head  of  cattle  at  $39  per 
head.     During  the  summer  97  died  of  disease,  but  he  sold  the 
remainder  so  as  to  gain  on  the  whole  number  $1819.     How 
much  did  he  get  for  his  cattle  per  head? 

21.  If  a  young  man  who   has  a  salary  of  $30  per  week, 
pays  $7.25  for  his  board  and  $4.25  for  other  expenses,  how 
long  will  it  take  him  to  save  $1500? 

22.  A  man  bought  a  horse  for  $115  and  after  keeping  him 
three  months,  sold  him  for  $155.     If  lie  paid  $30  for  his  keep- 
ing and  received  $50  for  the  use  of  him  during  that  time, 
how  much  did  he  gain? 

23.  A  speculator  purchased  a  certain  number  of  bushels 
of  wheat  for  $8735.     He  sold  it  for  $9215  and  in  so  doing 
gained  $  .25  per  bushel.     How  many  bushels  did  he  buy? 

24.  I  bought  25  barrels  of  flour  for  $200.     For  what  must 
it  be  sold  per  barrel  to  gain  $50?     What  will  be  the  gain  per 
barrel  ? 

25.  A  tailor  having  $585  wished  to  purchase  with  this  an 
equal  number  of  yards  of  two  kinds  of  broadcloth.     One  kind 
was  worth  $6  a  yard,  the  other  $7  a  yard.     How  many  yards 
of  each  kind  could  he  buy  ? 

26.  Two  men  leave  the  same  place  at  the  same  time  and 
travel  in  opposite  directions,  one  at  the  rate  of  48  miles  per 
day,  the  other  at  the  rate  of  52  miles  per  day.     How  far 
apart  will  they  be  at  the  end  of  5  days? 

27.  If  20  men  can  do  a  piece  of  work  in  31  days,  how  many 
days  will  be  required  to  do  an  equal  amount  of  work  if  11 
additional  men  are  employed? 

28.  A  farmer  wished  to  obtain  $120.     He  sold  16  barrels  of 
apples  at   $3.50  per  barrel,  and  enough  barley  at  $  .80  a 
bushel  to  make  up  the  sum  required.     How  many  bushels 
of  barley  did  he  sell? 


74  DIVISION. 

29.  Mr.  B.  bought  140  acres  of  land  for  $17500,  and  sold 
enough  at  $120  per  acre  to  amount  to  $9600.    The  rest  of  the 
land  he  sold  at  cost.    How  many  acres  did  he  sell  at  cost,  and 
what  was  the  entire  loss? 

30.  A  man  pays  $628  a  year  for  groceries,  $350  for  house 
rent,  $262  for  clothes,  twice  as  much  for  traveling  expenses 
as  for  house  rent,  $175  for  annual  premium  for  life  insurance, 
and  saves  in  4  years  enough  money  to  purchase  130  acres  of 
land  at  $53  an  acre.     What  is  his  yearly  income? 

31.  In  October,  1871,  the  great  fire  in  Chicago  burned  ovei 
an  area  of  2124  acres.     The  estimated  loss  occasioned  by  the 
fire  was  $196000000.     What  was  the  average  loss  per  acre? 

32.  A  boy  has  a  velocipede  which  he  can  run  at  the  rate 
of  140  rods  in  4  minutes.     How  many  minutes  will  it  take 
him  to  run  it  630  rods? 

33.  A  farmer  has  1000  head  of  cattle  in  5  fields.     In  the 
first  he  has  315  head,  in  the  second  175  head,  in  the  third 
300  head,  and  in  the  fourth  the  same  number  as  in  the  fifth. 
How  many  has  he  in  the  fifth  ? 

34.  A  man  gave  away  $45000  in  three  equal  amounts. 
One  share  he  gave  to  his  son,  one  share  to  his  daughter,  and 
the  rest  to   his  grandchildren,   giving  them   $1500  apiece. 
How  many  grandchildren  had  he  ? 

35.  In  the  Centennial  Exhibition,  at  Philadelphia,  a  section 
of  a  cable  in  process  of  construction  for  the  new  suspension 
bridge  at  New  York  was  shown.     It  was  composed  of  6000 
galvanized  steel  wires,  and  its  ultimate  strength  was  22,300,000 
pounds.     What  weight  was  each  wire  capable  of  sustaining  ? 

36.  The  main   building  of  the  Centennial  Exhibition   at 
Philadelphia,  the  largest  building  in  the  world,  contained  on  the 
ground  floor  an  area  of  872320  square  feet,  on  the  upper  floors 
in  projections  37344  square  feet,  in  towers  26344  square  feet. 
If  there  are  43560  square  feet  in  an  acre,  how  many  acres  did 
the  floors  of  the  building  contain  ? 


PROPERTIES  OF  NUMBERS 


108.    1.  What  is  the  product  of  4  times  5?     What  are  4 
and  5  of  their  product? 

2.  What  is  4  of  16?     Of  24?    What  is  7  of  14?     Of  28? 

3.  What  numbers  will  exactly  divide  18?     24?     36?     72? 

4.  Give  the  exact  divisors  of  42.     96.     108.     48.     32?. 

5.  What  are  the  factors  of  30?     24?     40?     56?     64? 

6.  What  numbers  between  0  and  10  can  not  be  divided  by 
any  number  except  themselves  and  1?     Between  10  and  20? 

7.  What  numbers  between  0  and  10  can  be  divided  by 
other  numbers  than  themselves  and  1?     Between  10  and  20? 


DEFINITIONS. 

109.  An  Integer  or  Integral  Number  is  one  that 
expresses  whole  units. 

Thus,  281,  36  houses,  46  men,  are  integral  numbers. 

110.  An  Exact  Divisor  of  a  number  is  an  integer  that 
will  divide  it  without  a  remainder. 

Thus,  2,  4,  6  and  12  are  exact  divisors  of  24. 

111.  The  Factors  of  a  number  are  the  integers  which 
being  multiplied  together  will  produce  the  number. 

Thus,  6  and  8  are  factors  of  48. 


The  exact  divisors  of  a  number  are  factors  of  it. 


(75) 


76  PROPERTIES   OF    NUMBERS. 

112.  A   Prime  Number   is  one  that  has  no  exact 
divisors  except  itself  and  1. 

Thus,  1,  3,  5  and  7  are  prime  numbers. 


L3.  A  Composite  Number  is  one  that  has  exact 
divisors  besides  itself  and  1. 

Thus,  18  and  24  are  composite  numbers,  for  18  is  divisible  by  6,  and 
24  by  8. 

111.  An  Even  Number  is  one  that  is  exactly  divisible 
by  2. 

Thus,  2,  4,  6,  8,  etc.,  are  even  numbers. 

115.  An  Odd  Number  is  one  that  is  not  exactly  divis- 
ible by  2. 

Thus,  1,  3,  5,  7,  9,  etc.,  are  odd  numbers. 

DIVISIBILITY  OF  NUMBERS. 

116.  In  determining  by  inspection  the  divisibility  of  num- 
bers, the  following  facts  will  be  found  valuable. 

1.  Two  is  an  exact  divisor  of  any  even  number. 
Thus,  2  is  an  exact  divisor  of  12,  16,  30  and  44. 

2.  Three  is  an  exact  divisor  of  any  number,  the  sum  of  whose 
digits  is  divisible  by  3. 

Thus,  3  is  an  exact  divisor  of  312,  135,  423,  and  3816. 

3.  Four  is  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  two  right  hand  figures  is  divisible  by  4. 

Thus,  4  is  an  exact  divisor  of  264,  1284,  1368,  and  7932. 

4.  Five  is  an  exact  divisor  of  any  number  whose  right  hand 
figure  is  0  or  5. 

Thus,  5  is  an  exact  divisor  of  360,  1795,  3810,  and  7895. 


DIVISIBILITY   OF    NUMBERS.  77 

5.  Six  is  an  exact  divisor  of  any  even  number,  the  sum  of 
whose  digits  is  divisible  by  3. 

Thus,  6  is  an  exact  divisor  of  732,  534,  798,  and  8226. 

6.  Eight  is  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  three  right  hand  figures  is  divisible  by  8. 

Thus,  8  is  an  exact  divisor  of  4328,  3856,  61360,  and  5920. 

7.  Nine  is  an  exact  divisor  of  any  number,  the  sum  of 
whose  digits  is  divisible  by  9. 

Thus,  9  is  an  exact  divisor  of  513,  1314,  252,  1341,  and  312462. 

8.  10,  100,  1000,  etc.,  are  exact  divisors  of  any  numbers 
that  end  respectively  with  one,  two,  three,  etc.,  ciphers. 

Thus,  10,  100,  1000,  etc.,  are  exact  divisors  respectively  of  80,  800, 
8000,  etc. 

9.  If  an  even  number  is  divisible  by  an  odd  number  it  is 
divisible  by  twice  that  number. 

Thus,  72  is  divisible  by  9  and  by  twice  9  or  18.     312  by  3  and  6. 

10.  An  exact  divisor  of  a  number  is  an  exact  divisor  of  any 
number  of  times  that  number. 

Thus,  3  is  an  exact  divisor  of  12,  and  of  any  number  of  times  12, 
as  36. 

11.  An  exact  divisor  of  each  of  two  numbers  is  an  exact 
divisor  of  their  sum  and  of  their  difference. 

Thus,  3  is  an  exact  divisor  of  9  and  12  respectively,  and  therefore 
of  9  +  12,  or  21 ;  of  12  —  9,  or  3. 

117.  Find  by  inspection  some  of  the  exact  divisors  of  the 
following  numbers: 


1.  1524. 

5.  2556. 

9.  42840. 

13.  376250. 

2.  3432. 

6.  7236. 

10.  92475. 

14.  428328. 

3.  4264. 

7.  27360. 

11.  362088. 

15.  4183200. 

4.  9360. 

8.  23661. 

12.  438408. 

16.  6853744. 

78  PROPERTIES   OF    NUMBERS. 


FACTORING. 

118.  1.  What  are  the  factors  of  6?     8?     12?     16? 

2.  What  factors  of  18  are  prime  numbers  or  prime  factors? 

3.  What  are  the  prime  factors  of  30? 

4.  What  are  all  the  exact  divisors  of  30? 

5.  What  numbers  besides  the  prime  factors  of  30  are  its 
exact  divisors?      How  are  they  obtained    from   the  prime 
factors  ? 

6.  Of  what  number  are  2,  3,  and  5,  the  prime  factors? 

7.  How  can  a  number  be  obtained  from  its  prime  factors? 

8.  The  prime  factors  of  a  number  are  2,  2,  and  5.     What 
is  the  number?    Give  all  the  exact  divisors  of  this  number. 

9.  What  are  the  exact  divisors  of  60?     72?     96?     144? 

DEFINITIONS. 

119.  Factoring  is  the  process  of  separating  a  number 
into  its  factors. 

120.  Prime   Factors  are   factors    that    are    prime 
numbers. 

121.  The  number  of  times  a  number  is  used  as  a  factor  is 
indicated  by  a  small  figure  called  an  exponent.     It  is  written 
above  and  at  the  right  of  the  number. 

Thus,  4  X  4  X  4  =  43,  and  the  3  indicates  that  4  is  used  as  a  factor 
three  times. 

122.  PRINCIPLES. — 1.  Every  prime  factor  of  a  number  is  an 
exact  divisor  of  that  number. 

2.   The  only  exact  divisors  of  a  number  are  its  prime  factors  or 
the  product  of  two  or  more  of  them. 

3-  Every  number  is  equal  to  the  product  of  its  prime  factors. 


FACTORING. 


79 


1.  What  are  the  prime  factors  of  756? 


PROCESS. 
2)756 

2)378 

3)189 

3)63 

3)21 

7 


ANALYSIS. — Since  every  prime  factor  of  a  number 
is  an  exact  divisor  of  the  number,  we  may  find  the 
prime  factors  of  756  by  finding  all  the  prime  numbers 
that  are  exact  divisors  of  756.  Since  the  number  is 
even,  we  divide  by  2.  Since  the  quotient  obtained  is 
an  even  number,  we  divide  again  by  2.  Then  we  di- 
vide by  the  prime  numbers  3,  3,  3,  successively,  and 
the  last  quotient  is  7,  which  is  a  prime  number. 

Hence  the  prime  factors  are  2,  2,  3,  3,  3,  7,  or  22,  33,  7. 


RULE. — Divide  the  given  number  by  any  prime  number  that 
will  exactly  divide  it.  Divide  this  quotient  by  another  prime  num- 
ber, and  so  continue  until  the  quotient  is  a  prime  number. 

The  several  divisors  and  last  quotient  will  be  the  prime  factors. 


What  are  the 

2.  Of    35? 

3.  Of    64? 

4.  Of  336? 

5.  Of  168? 

6.  Of  144? 

7.  Of  315? 

8.  Of  198? 

9.  Of  224? 

10.  Of  786? 

11.  Of  316? 


prime  factors 

12.  Of    484? 

13.  Of  1280? 

14.  Of  1008? 

15.  Of  1140? 

16.  Of  1184? 

17.  Of  1872? 

18.  Of  7644? 

19.  Of  2310? 

20.  Of  3204? 

21.  Of  4725? 


22.  Of 

23.  Of 

24.  Of 

25.  Of 

26.  Of 

27.  Of 

28.  Of 

29.  Of 

30.  Of 

31.  Of 


3913? 

3812? 

7007  ? 

3980? 
26840? 
38148? 
11340? 
24024? 
18500? 
124416? 


MULTIPLICATION  BY  FACTOES. 

123.  1.  What  are  the  factors  of  12?     16?     18?     20? 

2.  What  are  the  factors  of  24?     42?     36?     30?     27? 

3.  What  are  the  factors  of  45?     48?     56?     63?     72? 

4.  When  a  number  is  multiplied  by  4  and  the  product  by 
6,  by  what  is  the  number  multiplied? 


80  PROPERTIES   OF    NUMBERS. 

5.  What  will  20  carriages  cost  at  $346  each? 
PROCESS.  ANALYSIS.  —  Since  20  is  5  times  4,  20 

$  3  4  6  cost  of  1  carriage.       carriages  will  cost  5  times  as  much  as 
4  4   carriages.     4  carriages   will   cost  4 

times  $346,  or  $1384,  and  20  carriages 
>st  of  4  carriages.       wiR  ^  g  times  ^  mu^  ^  4  carriageR> 

or    5    times    $1384,    which   is    $6920. 
$  6  9  2  0  cost  of  20carriages.     Hence,  20  carriages  will  cost  $6920. 


RULE.  —  Multiply  the  multiplicand  by  one  factor  of  the 
plier,  the  product  thus  obtained  by  another  factor,  and  so  continue 
until  all  the  factors  have  been  used  successively  as  multipliers. 
The  last  product  will  be  the  product  sought. 

Multiply  in  same  manner,  using  the  factors  of  the  multiplier  : 

6.  425  by  32;  by  36;  by  48;  by  72. 

7.  1824  by  56;  by  27;  by  45;  by  108. 

8.  What  will  be  the  cost  of  35  cows  at  $64  each  ? 
9/  What  will  21  cords  of  wood  cost  at  $5.35  a  cord? 

10.  What  will  72  yoke  of  oxen  cost  at  $168  per  yoke? 

11.  What  will  36  boxes  of  lemons  cost  at  $6.25  per  box? 

12.  What  will  48  acres  of  land  cost  at  $46  per  acre? 

13.  What  will  24  paintings  cost  at  $55  each? 

14.  What  will  45  cases  of  boots  cost  at  $36  a  case  ? 

15.  What  will  56  barrels  of  salt  cost  at  $2.35  a  barrel? 

DIVISION  BY  FACTORS. 

124.    1.  What  are  the  factors  of  32?     25?     64?     96? 

2.  If  a  number  is  divided  by  8,  by  what  must  the  quotient 
be  divided  that  the  number  may  be  divided  by  16? 

3.  If  a  number  is  divided  by  8  and  the  quotient  by  6,  by 
what  is  the  number  divided? 

4.  What  factors  may  be  used  to  divide  a  number  by  36  ? 

5.  What  factors  may  be  used  to  divide  a  number  by  48  ? 


FACTORING. 


81 


6.  A  miller  put  tip  500  pounds  of  hominy  in  packages  con- 
taining 4  pounds  each,  and  packed  them  in  boxes  containing 
10   packages   each.     How   many  packages  and    how   many 
boxes  did  he  have  ? 

7,  Divide  888  by  24,  using  factors. 

ANALYSIS. — 24  is  equal  to  6  times  4.  Hence  to 
divide  by  24  we  may  divide  by  6  times  4.  888  -f-  4  = 
222.  But  since  we  were  to  divide  by  6  times  4,  this 
quotient  is  6  times  too  great,  hence  we  must  divide  it  by 
6.  222  ~  6  =  37  the  true  quotient. 


PROCESS. 

4)888 

6)222 

37 


8.  Divide  5863  by  32,  using  factors. 

PROCESS. 

4)5683 

3 


2)1420. 

4)710 
177, 


3  +  (2  X  8)  =  19  trueRem. 
1  7  7  if  Quotient. 

mainder  is  3  units,  and  the  second,  2  eights,  or  16  ;  hence  the  entire 
remainder  is  3  +  16,  or  19,  and  the  quotient  is  177-g-f. 


ANALYSIS. — 32  is  equal  to  4  X 
2  X  4.  Dividing  5683  by  4  gives 
a  quotient  of  1420  fours  and  3 
units  remaining. 

Dividing  1420  fours  by  2  gives 
a  quotient  ot  710  eights.  Dividing 
710  eights  by  4  gives  a  quotient 
of  177  thirty-twos  and  2  eights 
remainder.  The  first  partial  re- 


EULE. — Divide  the  dividend*  by  one  factor  of  the  divisor,  the 
quotient  thus  obtained  by  another  factor,  and  so  continue  until  all 
the  factors  have  been  used  successively  as  divisors. 

If  there  be  remainders,  multiply  each  remainder  by  all  the  preced- 
ing divisors  except  the  one  that  produced  it.  The  sum  of  these  prod- 
ucts ivill  be  the  true  remainder. 


17.  3275  by  56. 

18.  3276  by  27. 

19.  4104  by  45. 

20.  7304  by  24. 


Divide,  using  fact 

9.  1704  by  24. 
10.  4725  by  15. 
11.  5740  by  28. 

12.  1428  by  42. 
6 

ors: 

13.  1288  by  56. 
14.  3528  by  72. 
15.  3824  by  32. 
16.  2184  by  49. 

82  PROPERTIES   OF   NUMBERS. 

21.  A   wholesale   grocer   put  up   1120  pounds  of  tea  in 
35-pound    packages,    containing    5-pound    canisters.       How 
many  packages  and  canisters  were  there? 

22.  A  paper  manufacturer  put  up  his  paper  so  that  each 
quire  contained  4  packages  of  6  sheets  each.     How  many 
packages  and  quires  were  made  up  from  912  sheets? 


CANCELLATION. 

125.  1.  How  many  times  is  2  times  5  contained  in  4 
times  5  ?     2  times  7  in  4  times  7  ?     2  times  9  in  4  times  9  ? 
2  times  24  in  4  times  24?     2  times  any  number  in  4  times 
that  number? 

2.  How  many  times  is  4   times   8  contained  in  12  times 
8?     4  times  25  in  12  times  25?     4  times  75  in  12  times  75? 
4  times  any  number  in  12  times  the  same  number? 

3.  How  many  times  is   6  X  48    contained    in   24  X  48  ? 
6X  144  in  24  X  144? 

4.  In  determining  the   quotient,  what  numbers   may  be 
omitted  from  both  dividend  and  divisor? 

126.  Cancellation  is  the  process  of  shortening  compu- 
tations by  rejecting  equal  factors  from  the  dividend  and  divisor. 

127.  PRINCIPLE. — Rejecting  equal  factors  from  both  dividend 
and  divisor  does  not  alter  the  value  of  the  quotient. 

1.  Divide  66  times  36  by  24  times  11. 

PROCESS.  ANALYSIS. — We  write 

66X36       ^X^X^X3X3  the  numbers  as  in  divis- 

—  =  -—  —  —  9     ion,  the  dividend  above. 

24X11         0X^X;j  the  divisor  below  a  line. 

Instead  of  multiplying  66  by  36  we  resolve  66  into  its  factors  11  and 
6,  and  36  into  its  factors  4,  3  and  3,  and  in  the  divisor  resolve  24  into 
the  factors  6  and  4. 


CANCELLATION.  83 

Cancelling  equal  factors  from  both  dividend  and  divisor,  which  is 
the  same  as  dividing  both  by  the  same  number,  and  does  not  alter 
the  value  of  the  quotient,  we  have  remaining  in  the  dividend  the 
factors  3  and  3,  or  9,  which  is  the  quotient. 

2.  Divide  72  X  66  X  49  by  63  X  40  X  21. 

PROCESS.  ANALYSIS. — We  write  the  num- 

g         22        y  kers  as  before.     Since  9  is  a  factor 

!72V$$V4$       22  °^  k°tn  72  and  63  it  may  be  rejected 

*  =  —  =  4f .     from  both,  leaving  8  instead  of  72  in 
0  jjTX '£  j*  X  $?         5  the  dividend,  and  7  instead  of  63  in 

/f  5          fl  the  divisor.     We  next  cancel  8  from 

8  and  40,  leaving  5  instead  of  40  in  the  divisor.  We  next  cancel  7 
from  7  and  49,  leaving  7  instead  of  49  in  the  dividend,  and  7  again 
from  7  and  21,  leaving  3  instead  of  21.  Rejecting  the  factor  3  from 
both  66  and  3,  there  is  left  for  a  dividend  22,  and  for  a  divisor  5,  which 
gives  a  quotient  of  4f . 

RULE. — Reject  from  the  dividend  and  divisor  all  factors  common 
to  both9  and  then  divide  the  product  of  the  remaining  factors  of  the 
dividend  by  the  product  of  the  remaining  factors  of  the  divisor. 

When  all  the  factors  of  both  dividend  and  divisor  are  cancelled,  the 
quotient  is  1,  for  the  dividend  will  then  exactly  contain  the  divisor 
once. 

EXAMPLES. 

Divide,  using -cancellation : 

3.  7  X  5  X  3  X  11  by  5  X  11  X  3. 

4.  12  X  14  by  6  X  7  X  2. 

5.  6  X  3  X  5  X  2  by  3  X  5  X  2  X  2. 

6.  4  X  2  X  8  X  24  by  36  X  8  X  2. 

7.  24  X  32  by  8  X  6  X  4. 

8.  45  X  60  X  7  by  49  X  12  X  9. 

9.  2X3X5X8X7  by  6X5X2X7. 

10.  5  X  8  X  12  X  6  by  20  X  16  X  2. 

11.  12  X  60  X  36  X  35  by  7  X  30  X  18  X  24. 

12.  30  X  49  X  64  X  25  by  35  X  15  X  24. 


84  PKOPERTIES    OF    NUMBERS. 

13.  Divide   the   product  of  26  times  18  times  35,  by  78 
times  30. 

14.  Find  the  quotient  of  99  times  360  times  365,  divided 
by  11  times  72. 

15.  Find  the  quotient  of  175  X  28  X  72  times  363,  divided 
by  12  X  11  X  9. 

16.  Four  farms  containing  80  acres  each,  worth  $65  per  acre, 
were  exchanged  for  5  farms  containing  95  acres  each.    What 
was  the  value  per  acre  of  the  farms  received  in  exchange? 

17.  A  farmer  buys  3  pieces  of  muslin  each  containing  44 
yards  at  11  cents  a  yard,  and  pays  for  it  in  wheat  at  $2  per 
bushel.     How  many  bushels  are  required? 

18.  A  merchant  bought  13  tubs  of  butter,  each  containing 
39  pounds,  at  32  cents  a  pound,  paying  for  it  in  4  patterns  of 
silk  of  13  yards  each.     How  much  was  the  silk  a  yard? 


COMMON  DIVISORS. 

128.    1.  What  numbers  will  exactly  divide  12?     15?     20? 

2.  What  numbers  will  exactly  divide  both  12  and  15?     15 
and  20?     24  and  48?     63  and  72? 

3.  What  numbers  will  exactly  divide  both   12   and   24? 
What  is  the  largest  number  that  will  exactly 'divide  them? 

4.  What  is  the  largest  number  that  will  exactly  divide 
both  15  and  30?     16  and  32?     16  and  24?     24  and  32? 

5.  Name  all  the  divisors  common  to  15  and  30. 

6.  Name  all  the  prime  divisors  or  factors  common  to  15 
and  30? 

7.  How  is  the  greatest  divisor  common  to  15  and  30  found 
from  the  prime  factors  of  those  numbers? 

8.  What  is  the  greatest  divisor  common  to  24  and  30? 

9.  How   is    the  greatest   divisor  common   to   24  and   30 
obtained  from  the  prime  factors  of  those  numbers  ? 


COMMON    DIVISORS.  85 


DEFINITIONS. 

129.  A  Common  Divisor  of  two  or  more  numbers 
is  an  exact  divisor  of  each  of  them. 

Thus,  6  is  a  common  divisor  of  12,  24,  48  ;  8  of  16,  24  and  64. 

130.  The  Greatest  Common  Divisor  of  two  or 

more  numbers  is  the  greatest  number  that  is  an  exact  divisor 
of  each  of  them. 

Thus,  24  is  the  greatest  common  divisor  of  24  and  48. 

131.  When  numbers  have  no  common  divisor  they  are  said 
to  be  Prime  to  each  other. 

Thus,  7,  8  and  9  are  prime  to  each  other. 

A  common  divisor  is  sometimes  called  a  common  measure  and 
the  greatest  common  divisor  the  greatest  common  measure. 

132.  PRINCIPLE.  —  The  greatest  common  divisor  of  two  or  more 
numbers  is  the  product  of  all  their  common  prime  factors. 

1.  What  is  the  greatest  common  divisor  of  45,  60,  and  75? 

IST  PROCESS.  ANALYSIS.  —  Since  the   greatest  coni- 

45  =  3v3v5  mon  divisor  is  equal  to  the  product  of 

n  f\  _  c>  \/  c>  \/  €>  ^/  K     all  the    prime  factors   common  to  the 

O  U  —  L  X  -A  X  d  X  0  i  ^i 

^  given  numbers,  we   separate  the  num- 

'  **  =  ^  '  bers  into  their  prime  factors.     The  only 

3  X  5  =  15  prime  factors  common  to  all  these  num- 

bers are  3  and  5.     Hence  their  product, 

15,  is  the  greatest  common  divisor  of  the  given  numbers. 

2o  PROCESS.  ANALYSIS.  —  3  will  divide  each  of  the 

3145       60       75          given  numbers,  and  is  therefore  a  factor 
—-^  -  —  -  —          of  the  greatest  common  divisor.    5  will 
divide  each  of   the  resulting  quotients 


3           4         .5          and  is  therefore  a  factor  of  the  greatest 
3  X  5  =  15  common    divisor.     The   quotients    3,   4, 

and  5,  have  no  common  divisor;  there- 
fore 3  and  5  are  the  only  factors  of  the  greatest  common  divisor,  15. 


86 


PROPERTIES    OF    NUMBERS. 


RULE. — Separate  the  numbers  into  their  prime  factors  and  find 
the  product  of  all  the  common  factors.  Or, 

Divide  the  numbers  by  any  common  divisor,  the  resulti7ig  quo- 
tients by  another  common  divisor,  and  so  continue  to  divide  until 
quotients  are  obtained  that  have  no  common  divisor. 

The  product  of  the  divisors  will  be  the  greatest  common  divisor. 


EXAMPLES. 

What  is  the  greatest  common  divisor  of 

2.  12,  16,  20,  24? 

3.  18,  27,  36,  45? 

4.  24,  48,  60,  72? 

5.  36,  60,  72,  66? 

6.  48,  72,  96,  84? 

7.  18,  81,  72,  54? 

8.  32,  48,  80,  96? 

9.  45,  63,  99,  81  ? 
10.  35,  56,  84,  63? 

133.  When  the  numbers  can  not  be  factored  readily,  the 
following  method  is  employed: 

1.  What  is  the  greatest  common  divisor  of  35  and  168  ? 

PROCESS.  ANALYSIS. — The   greatest  common 


11. 

12. 

13. 
14. 
15. 
16. 
17. 
18. 
19. 

16, 
36, 
30, 
30, 
14, 
24, 
33, 
24, 
42, 

40, 
60, 

55, 
54, 
42, 

28, 
77, 
72, 
84, 

72; 

84, 
85, 
66, 
63, 
120, 
143,- 
120, 
252, 

88? 
96? 
90? 
78? 
91? 
144? 
154? 
168? 
294? 

35)168(4 
140 

28)35(1 
28 


divisor  can  not  be  greater  than  the 
smaller  number;  therefore  35  will  be 
the  greatest  common  divisor  if  it  is 
exactly  contained  in  168.  By  trial  it 

is  found  that  it  is  not  an  exact  divisor 

7)28(4      °^  168,  since  there  is  a  remainder  of 
2  Q  28.    Therefore  35  is   not  the  greatest 

common  divisor. 

Since  168  and  140,  which  is  4  times  35,  are  each  divisible  by  the 
greatest  common  divisor,  their  difference,  28,  must  contain  the  greatest 
common  divisor;  therefore  the  greatest  common  divisor  can  not  be 


COMMON  BIVISOES.  87 

greater  than  28.  28  will  be  the  greatest  common  divisor  if  it  is  exactly 
contained  in  35;  since  if  it  be  contained  in  35,  it  will  be  contained  in 
140,  and  in  28  plus  140,  or  168.  By  trial  we  find  that  it  is  not  an  exact 
divisor  of  35,  for  there  is  a  remainder  of  7.  Therefore  28  is  not  the 
greatest  common  divisor. 

Since  28  and  35  are  each  divisible  by  the  greatest  common  divisor, 
their  difference,  7,  must  contain  the  greatest  common  divisor;  therefore 
the  greatest  common  divisor  can  not  be  greater  than  7.  7  will  be  the 
greatest  common  divisor  if  it  is  exactly  contained  in  28 ;  since  if  it  be 
contained  in  itself  and  28,  it  will  be  contained  in  their  sum,  35,  and  also 
in  168,  which  is  the  sum  of  28  and  4  times  35,  or  140.  By  trial  we  find 
that  it  is  an  exact  divisor  of  28.  Hence  7  is  the  greatest  common  divisor. 

RULE. — Divide  the  greater  number  by  the  less  and  if  there  be  a 
remainder  divide  the  less  number  by  it,  then  the  preceding  divisor 
by  the  last  remainder,  and  so  on,  till  nothing  remains.  The  last 
divisor  will  be  the  greatest  common  divisor. 

If  more  than  two  numbers  are  given,  find  the  greatest  common 
divisor  of  any  two,  then  of  this  divisor  and  another  of  the  given  num- 
bers, and  so  on.  The  last  divisor  will  be  the  greatest  common  divisor. 

Find  the  greatest  common  divisor  of 


2.  169  and  195. 

3.  187  and  209. 

4.  372  and  492. 

5.  119  and  187. 

6.  243  and  297. 

7.  322  and  391. 


8.  252  and  294. 

9.  156  and  208. 

10.  702  and  945. 

11.  1029  and  1197. 

12.  1666  and  1938. 

13.  3596  and  3768. 


What  is  the  greatest  common  divisor  of 


14.  672,  352,    992? 

15.  714,  867,  1088? 

16.  462,  759,  1155? 


17.  630,  1134,  1386? 

18.  462,  1764,  2562? 

19.  7955,  8769,  6401  ? 


20.  In  a  village  some  of  the  walks  are  56  inches  wide, 
some  70  inches,  and  others  84  inches.  What  is  the  width  of 
the  widest  flagging  that  will  suit  all  the  walks? 


88  PROPERTIES    OF    NUMBERS. 

21.  A  merchant  has  60  pounds  of  tea  of  one  kind,  75 
pounds  of  another,   and   100  pounds   of  another,   which  he 
wishes  to  put  up  in  the  largest  possible  equal  packages  with- 
out mixing  the  different  kinds.     How  many  pounds  should  be 
put  in  each  package? 

22.  Mr.  A.  has  324  acres  of  land  in  one  farm  and  78  acres  in 
another.     He  wishes  to  divide  these  into  the  largest  possible 
fields  of  equal  size.     How  many  fields  will  there  be,  and  how 
many  acres  in  each  field? 

MULTIPLES. 

134.  1.  What  numbers  less  than  25  will  exactly  contain 
4?     5?     6? 

2.  What  numbers  less  than  25  will  exactly  contain  both  4 
and  6? 

3.  Name  some   numbers  that  are   exactly  divisible  by  5. 
By  4.     By  both  5  and  4. 

4.  Name  some   numbers  that  are  exactly  divisible  by  2. 
By  3.     By  4.     By  2  and  4. 

5.  What  is  the  smallest  number  that  is  exactly  divisible  by 
each  of  the  numbers  2,  3,  and  4? 

6.  What  is  the  least  number  that  will  contain  10  and  15? 

7.  What  common  prime  factors  have  10  and  15?     What 
factor  occurs  in  10  that  does  not  in  15?    What  factor  is  found 
in  15  that  is  not  found  in  10? 

8.  AVhat  are  all  the  different  prime  factors  of  10  and  15? 

9.  How  may  the  least  number  that  will  contain  10  and  15 
be  formed  from  their  prime  factors? 

What  is  the  least  number  that  will  exactly  contain 


10.  3,  6  and    9? 

11.  3,  5  and    6? 

12.  4,  8  and  12? 


13.  2,  3,  5  and    6? 

14.  3,  4,  5  and    6? 

15.  3,  6,  8  and  12? 


MULTIPLES.  89 


DEFINITIONS. 

135.  A  Multiple  of  a    number  is  a  number  that  will 
exactly  contain  it. 

A  multiple  of  a  number  is  obtained  by  multiplying  the  given  number 
by  some  integer. 

136.  A  Common  Multiple  of  two  or  more  numbers, 
is  a  number  that  will  exactly  contain  each  of  them. 

137.  The   Least   Common  Multiple  of  two  or 

more  numbers,  is  the  least  number  that  will  exactly  contain 
each  of  them. 

138.  PRINCIPLE. — The  least  common  multiple  of  two  or  more 
numbers  is  equal  to  the  product  of  all  the  prime  factors  of  the 
numbers,  and  no  other  factors. 

WRITTEN    EXERCISES. 

139.  1.  Find  the  least  common  multiple  of  30,  28  and  60? 

IST  PROCESS.  ANALYSIS. — Since  the  least  com- 

OQ 2  Y  3  "X  5  mon  multiple  is  equal  to  the  product 

~  Q 9  \/  9  v  7  of  all  the  different  prime  factors  of 

the   numbers   and  no  other  factors, 
(Prin.)  the  numbers  must  be  sepa- 

2  X2X3X5X7— 420      rated   into  their  prime  factors,  and 

the  product  of  all  the  different  prime 
factors  found.     The  prime  factors  of  60,  the  largest  number,  are  2,  2, 

3  and  5.     28  contains  a  factor,  7,  which  is  not  found  in  60.    60  con- 
tains  all  the  factors  of  the  other    number,  30.      Therefore   all    the 
different  prime  factors  of  the  given   numbers   are  2,   2,   3,  5  and  7, 
and  their  product,  420,  is  the  least  common  multiple. 

RULE. — Separate  the  given  numbers  into  their  prime  factors. 

Find  the  product  of  all  the  different  prime  factors,  using  each 
factor  the  greatest  number  of  times  it  occurs  in  any  of  the  given 
numbers. 


90 


PROPERTIES    OF    NUMBERS. 


Find  the  least  common  multiple  of 


2.  28,  32  and  64. 

3.  36,  72  and  144. 

4.  45,  70  and  90. 


5.  12,  16,  18  and  24. 

6.  15,  20,  25  and  30. 

7.  18,  54,  90  and  180. 


What  is  the  least  common  multiple  of 


8.  22,  55  and  77? 

9.  48,  60  and  180? 

10.  10,  64  and  96? 

11.  81,  63  and  135? 

12.  25,  70  and  95? 


13.  12,  18,  24  and  96? 

14.  32,  56,  64  and  80? 

15.  14,  35,  50  and  28? 

16.  33,  99,  84  and  135? 

17.  17,  51,  65  and  121? 


PROCESS. 


2 
2 
5 

16 

20  ' 

•  30 

8 

10 

15 

4 

5 

15 

4 

1 

3 

2X2X5X4X3  = 


18.  Find  the  least  common  multiple  of  16,  20,  and  30. 

ANALYSIS. — Since  2  is  a  prime  fac- 
tor of  each  of  the  numbers,  it  is  also  a 
factor  of  the  least  common  multiple. 
(Prin.)  Dividing,  there  remain  as 
the  other  factors  of  the  numbers,  8, 
10,  and  15.  2  is  a  prime  factor  of  8 
and  10,  and  is  therefore  a  factor  of 
the  least  common  multiple.  Divid- 
ing, there  remain  4,  5,  and  15. 

5  is  a  prime  factor  of  5  and  15,  and  is  therefore  another  factor  of 
the  least  common  multiple.  Dividing,  there  remain  4,  1,  and  3,  which 
are  prime  to  each  other.  Therefore  the  product  of  the  factors  2,  2,  5, 
4,  and  3,  will  be  the  least  common  multiple. 

RULE. —  Write  the  given  numbers  in  a  horizontal  line.  Divide 
by  any  prime  number  that  is  an  exact  divisor  of  two  or  more  of  the 
given  numbers,  and  write  the  quotients  and  undivided  numbers  in 
a  line  beneath. 

Thus  continue  to  divide  until  the  quotients  and  undivided  num- 
bers are  prime  to  each  other.  The  product  of  the  divisors,  and 
the  numbers  in  the  last  horizontal  line,  will  be  the  least  common 
multiple. 


MULTIPLES.  91 

In  finding  the  least  common  multiple,  all  numbers  that  are  factors 
of  other  given  numbers  may  be  disregarded.  Thus,  the  multiples  of 
8, 16,  32,  64,  80,  and  128  are  the  same  as  the  multiples  of  80  and  128. 


EXAMPLES. 


Find  the  least  common  multiple  of 


19.  60,  40,  120  and  72. 

20.  81,  45,  108  and  135. 

21.  40,  60,    80  and  120. 


22.  32,  36,  72  and  80. 

23.  30,  75,  60  and  90. 

24.  24,  44,  65  and  100. 


What  is  the  least  common  multiple  of 


25.  8, 12, 16,  24  and  48? 

26.  16,  20,  24,  32  and  40? 


27.  25,  40,  75,  80  and  120? 

28.  32,  45,  70,  64  and    90? 


29.  What  is  the  smallest  number  that  will  exactly  contain 
16,  24,  and  30? 

30.  How  long  must  a  box  be  that  no  room  may  be  lost  in 
packing  in  it  books  6  inches,  8  inches,  or  12  inches  long? 

31.  A  lady  desires  to  purchase  a  piece  of  cloth  that  can 
be  cut  without  waste,  into  parts  4,  5,  or  6  yards  long.     How 
many  yards  must  the  piece  contain? 

32.  I  have  a  certain  number  of  pennies  which  I  can  ar- 
range in  either  4,  6,  8,  10,  or  12  equal  piles.     What  num- 
ber of  pennies  have  I,  if  it  is  the  least  number  that  admits 
of  such  arrangement? 

33.  How  many  bushels  will  the  smallest  bin  contain  that 
can  be  emptied  by  taking  out  either  7  bushels,  10  bushels,  or 
30  bushels  at  a  time  ? 

34.  Four  agents  start  from  New  York  at  the  same  time. 
The  first  makes  his  trip  in  8  weeks,  the  second  in  9  weeks, 
the  third  in  15  weeks,  and  the  fourth  in  20  weeks.     How 
many  weeks  will  pass  by  before  they  will  again  start  out  from 
New  York  together  ? 


92  PROPERTIES    OF    NUMBERS. 

35.  Three  men  walk  around  a  circular  island,  the  circum- 
ference of  which  is  360  miles.     A  \valks  15  miles  a  day,  B 
18  miles  a  day,  and  C  24  miles  a  day.     If  they  start  together 
and  walk  in  the  same  direction,  how  many  days  will  elapse 
before  they  will  be  together  again  ? 

36.  Divide  5  X  15  X  80  X  56  X  81  by  10  X  5  X  16  X  78. 

37.  If  a  man  buys  a  lot  whose  sides  measure  respectively 
48  feet,  60  feet,  96  feet  and  108  feet,  what  will  be  the  length 
of  the  longest  boards  which  he  can  use  to  fence  all  the  sides 
without  cutting? 

38.  Find  the  greatest  common  divisor  of  1744,  9564  and 
8524. 

39.  What  is  the  smallest  number  which  can  be  divided  by 
250,  350,  and  525  respectively,  and  leave  a  remainder  of  25  ? 

40.  What  is  the  greatest  common  divisor  of  1728,  6912, 
and  8640? 

41.  A  stock  buyer  wishes  to  invest  the  same  amount  of 
money  in  sheep  at  $3  each,  hogs  at  814  each,  and  cows  at  $21 
each,  as  he  does  in   beef  cattle  at  $48  each.     What  is  the 
smallest  possible  amount  which  he  can  invest  in  each  ? 

42.  Jones  Brothers  &  Co.,  of  Cincinnati,  O.,  received  an 
order  for  a  number  of  Lyman's  Historical  Charts.     It  was 
found  that  if  the  charts  were  packed  in  boxes  containing 
either  24,  28,  32,  or  36  charts  each,  there  was  a  remainder 
of  9  each  time,  but  if  packed  in  boxes  containing  25  each, 
there  was  no  remainder.     How  many  charts  were  ordered? 

43.  A  dealer  in  real  estate  purchased  3  lots  of  land  whose 
width  on  the  street  were  respectively  152  rods,  288  rods,  and 
184  rods.     What  is  the  width  of  the  largest  lots  of  equal  size 
which  can  be  formed  from  them  ? 

44.  Divide  3  X  5  X  20  X  10  X  3X  13  by  26  X  9  X  3  X  4. 

45.  Find  the  greatest  common  divisor  of  2219,  4501,  and 
5964. 

46.  Divide  5x8x3x7X  28  X  99  by  11x4x7x5x4. 


140.  1.  When  a  line  is 
divided  into  two  equal  parts, 
what  is  each  part  called? 

2.  When  a  line  is  divided 
into  three  equal  parts,  what 
is  each  part  called?      What 
are  two  of  the  parts  called? 

3.  When  a  line  is  divided 
into  four  equals  parts,  what 
is  each  part  called?     What 
are  two  of  the  parts  called  ? 
called? 


one- 
fifth 


What  are  three  of  the  parts 


4.  When   any   thing   is    divided    into   five    equal    parts, 
what  is   each   part  called?     What  are  three   parts   called? 
What  are  four  parts  called? 

5.  When  things  are  divided  into  6,  7,  8,  9,  10,  15  equal 
parts  respectively,  what  is  each  of  the  parts  called?     What 
are  four  of  them  called  ? 

6.  How  many  halves  are  there  in  any  thing?     How  many 
thirds?     Fourths?     Fifths?     Sixths?     Tenths? 

7.  If  10  marbles  are  separated  into  5  equal  groups,  what 
part  of  the  marbles  will  be  in  each  group  ? 

8.  How  many  are  one-fifth  of  10?      Two-fifths?     Three- 
fifths? 

9.  How  many  are  one-sixth  of  12?     Two-sixths? 

(93) 


94  COMMON    FBACTIONS. 


DEFINITIONS. 

141.  A   Fraction  is  one  or  more  of  the  equal  parts 
of  a  unit. 

142.  The  Unit  of  a  Fraction  is  the  unit  which  is 
divided  into  equal  parts. 

A  fraction  in  which  the  unit  has  been  divided  into  any  number  of 
equal  parts  is  called  a  Common  Fraction. 

A  fraction  in  which  the  unit  has  been  divided  into  tenths,  hun- 
dredths,  thousandths,  etc.,  is  called  a  Decimal  Fraction. 

143.  A  Fractional  Unit  is  one  of  the  equal  parts  into 
which  a  unit  is  divided. 

144.  Since  a  fraction  is  one  or  more  of  the  equal  parts  of 
any  thing,  to  express  a  fraction  two  numbers  are  necessary, 
one  to  express  the  number  of  equal  parts  into  which  the  unit 
has  been  divided,  the  other  to  express  how  many  make  the 
fraction.      These  numbers  are  written  one  above  the  other 
with  a  horizontal  line  between  them. 

145.  The  Denominator  is  the  number  which  shows 
into  how  many  equal  parts  the  unit  is  divided. 

It  is  written  below  the  line. 

Thus,  in  the  fraction  f ,  7  is  the  denominator.  It  shows  that  the 
unit  of  the  fraction  has  been  divided  into  7  equal  parts. 

146.  The  Numerator  is  the  number  which  shows  how 
many  fractional  units  form  the  fraction. 

It  is  written  above  the  line. 

Thus,  in  the  fraction  \ ,  5  is  the  numerator  and  shows  how  many 
fractional  units  form  the  fraction. 

147.  The    numerator    and    denominator    are    called    the 
Terms  of  a  Fraction. 


COMMON    FRACTIONS.  95 

148.  Fractional  units  are  named  from  the  number  of  parts 
into  which  the  unit  is  divided.     Thus,  ^  is  read  one-sixth; 
^,  one-seventh. 

Fractions  are  read  by  naming  the  number  and  kind  of  frac- 
tional units.  Thus,  -f  is  read  five-sixths ;  ^,  five  twenty-firsts ; 
•g-f ,  thirteen  thirty-fifths. 

149.  Read  the  following: 

TB~  TT          TT5"          ¥t 


325  829  469,          385 

639          "3~T5"         "83T          986 

Express  by  figures: 

1.  Three   elevenths.     Five    thirteenths.      Eight   twenty- 
firsts. 

2.  Forty-eight  fiftieths.     Twenty-seven  eighty-fifths. 

3.  Sixty  forty-eighths.     Fifty-seven  ninety-ninths. 

4.  Forty-two  eighty-sevenths.     Thirty-nine  ninety-thirds. 

5.  Seventy-four  one-hundredths.      Ninety-seven  one-hun- 
dred-fifths. 

6.  Fifty-two  seventy-eighths.     Thirty-six  eighty-fourths. 

7.  Two  hundred  three-hundred-ninetieths. 

8.  Seven  hundred  seventy-one  eight-hundred-sixtieths. 

9.  Two  hundred  forty-nine  three-hundredths. 

10.  Five  hundred  sixty-six  seven-hundred-fiftieths. 

11.  One  hundred  eleven  two-hundredths. 

12.  Four  thousand  six  hundred  thirty  five-thousandths. 

Fractions  are  classified  with  reference   to  the  relation  of 
numerator  and  denominator  thus: 

150.  A  Proper  Fraction  is  one  in  which  the  numer- 
ator is  less  than  the  denominator. 

Thus,  },  f,  -|f ,  etc.,  are  proper  fractions. 

The  value  of  a  proper  fraction  is  therefore  less  than  1. 


96  COMMON    FRACTIONS. 

151.  An  Improper  Fraction  is  a  fraction  in  which 
the  numerator  equals  or  exceeds  the  denominator. 

Thus,  f,  |,  |f,  are  improper  fractions. 

The  value  of  an  improper  fraction  is  therefore  1  or  more  than  1. 

152.  A  Mixed  Number  is  a  number  expressed  by  an 
integer  and  a  fraction. 

Thus,  2 1,  5J,  are  mixed  numbers. 

Mixed  numbers  are  read  by  naming  the  fraction  after  the 
whole  number.  Thus,  2f  is  read  two  and  three-fourths. 

Fractions  may  be  regarded  as  expressing  unexecuted  divis- 
ion. Thus,  -ig6:  is  equal  to  16  -f-  8  ;  \5-  is  read  15-7-3. 

153.  1.  Interpret  the  expression  -f-. 

ANALYSIS. — f  represents  5  of  7  equal  parts  into  which  any  thing  is 
divided.  It  also  represents  one-seventh  of  five,  and  5  divided  by  7. 
It  is  read  five-sevenths. 

In  like  manner  interpret: 


2.    f 

5. 

if 

8. 

196' 

11. 

3.  -jSp 

6. 

If 

9. 

AV 

12. 

4-  A- 

7. 

T^- 

10. 

iff- 

13. 

REDUCTION. 

CASE    I. 
154.  To  reduce  fractions  to  larger,  or  higher  terms. 

1.  In  -|-  of  an  apple  how  many  fourths  are  there?     How 
many  eighths  ? 

2.  How  many  sixths  are  there  in  £?     How  many  ninths? 
How  are  the  terms  of  the  fraction  f  obtained  from  those  of 
l?     ffromi? 

3.  How  many  eighths  are  there  in  J?    How  many  twelfths? 


REDUCTION.  97 

4.  How  do  the  terms  of  the  fraction  -f  compare  with  the 
terms  of  the  fraction  ^? 

5.  In  what  equivalent  fraction  can  -J-  be  expressed  ? 

6.  How  do  the  terms  of  the   fraction   -J-   compare  with 
those  of -j^? 

7.  How  are  the  terms  of  the  fraction  -^  obtained  from 
those  of  £? 

8.  How  are  the  terms  of  the  fraction  f  obtained  from  ^? 

9.  How  are  the  terms  of  the  fraction  f  obtained  from  |-? 
10.  What  then  may  be  done   to  the  terms  of  a  fraction 

without  changing  "the  value  of  the  fraction  ? 


11.  Change  |  to  24ths. 

12.  Change  f  to  16ths. 

13.  Change  f  to  24ths. 

14.  Change  f  to  12ths. 


15.  Change  -&  to  36ths. 

16.  Change  f  to  20ths. 

17.  Change  %  to  14ths. 

18.  Change  f  to  18ths. 


155.  Reduction  of  Fractions  is  the  process  of 
changing  their  form  without  changing  their  value. 

156.  A  fraction  is  expressed  in  Larger  or   Higher 

Terms  when  its  numerator  and  denominator  are  expressed 
by  larger  numbers. 

157.  PRINCIPLE.  —  Multiplying  both  terms  of  afraetion  by  the 
same  number,  does  not  change  the  value  of  the  fraction. 

WRITTEN    EXERCISES. 

1.  Change  ^  to  forty-fifths. 

PROCESS.  ANALYSIS.  —  Since  there  are  45  forty-fifths  in 

4  5  -f-  1  5  ^=.  3         1,  in  T^  there  are  3  forty  -fifths;  and  in  T7^  there 


7   v  q  —  91 

Since  the  denominator  of  the  required  frac- 

15  X  3  =  45  tion  is  3  times  that  of  the  given  fraction,  we 

must  multiply  the  terms  of  the  fraction  by  3. 


98  COMMON    FRACTIONS. 

RULE. — Multiply  the  terms  of  the  fraction  by  such  a  number  as 
will  change  the  given  denominator  to  the  required  denominator. 


Reduce : 


2.  |f  to  50ths. 

3.  U  to  60ths. 

4.  |f  to  70ths. 

5.  ||  to  84ths. 

6.  £f  to  40ths. 

7.  H  to  54ths. 

8.  ||  to  66ths. 

9.  ff  to  54ths. 
10.  -      to  84ths. 


Reduce : 


11.  If  to  120ths. 

12.  f|  to  64ths. 

13.  ||  to  74ths. 

14.  ff  to  210ths. 

15.  |f  to  240ths. 

16.  ff  to  225ths. 

17.  f|  to  348ths. 

18.  f|  to  558ths. 

19.  f  f  to  235ths. 


CASE    II. 
158.  To  reduce  fractious  to  smaller,  or  lower  terms. 

1.  How  many  fourths  are  there  in  |?     How  many  in  ^? 

2.  How  many  thirds  are  there  in  |?     How  many  in  ^? 

3.  How  does  the  number  of  eighilis  of  any  thing  compare 
with  the  fourths?    Thirds  with  sixths?     Halves  with'  eights? 

4.  How  do  the  terms  of  the  fraction  ^  compare  with  those 
of  |  ?     How  with  those  of  ^? 

5.  How  do  the  terms  of  the  fraction  |  compare  with  those 
of  |  ?    How  with  those  of  •&? 

6.  How  are  the  terms  of  the  fraction  ^  obtained  from  those 
of  the  fraction  |?     How  from  those  of  ^-? 

7.  How  are  the  terms  of  the  faction  |  obtained  from  f  ? 

8.  What  then  may  be  done  to  the  terms  of  a  fraction 
without  changing  the  value  of  the  fraction? 

9.  Express  TV;   -^-,    •&,  in  smaller  or  lower  terms. 

10.  Express  -i-f ,   -|f ,    |4>  ^n  smaller  or  lower  terms. 

11.  Reduce  ^-,   £f,    f|,   to  smaller  or  lower  terms. 

12.  Reduce  ff ,  -j^j-,  -j^,  to  smaller  or  lower  terms. 


REDUCTION.  99 

159.  A  fraction  is  expressed  in  Smaller 9  or  Lower 
Terms  \\hon  its  numerator  and  denominator  are  expressed 
in  smaller  numbers. 

160.  A    fraction    is    expressed    in    the    Smallest,    Or 
JjOtrest  Terms  when   its   numerator   and  denominator 
have  no  common  divisor. 

161.  PRINCIPLE. — Dividing  both  terms  of  a  fraction  by  the 
same  member  does  not  change  the  value  of  the  fraction. 


WRITTEN    EXERCISES. 

162.    1.  Change  -||  to  an  equivalent  fraction  expressed  in 
its  smallest,  or  lowest  terms. 

PROCESS.  ANALYSIS. — Since   the   denominator  of 


32  _4 

48  ~4 


8         2         the  required  fraction  is  to  be  smaller  than 
7^  =  17        that  of  the  given  fraction,  we  may  obtain 


an    equivalent    fraction    having    smaller 
Or,  terms,  by  dividing  the  terms  of  the  given 

Q9       Q9  ••    1fi       9       fraction  by  any  exact  divisor,  as  4  (Prin.), 

—  — '. —  —      and  the  terms  of  the  resulting  fraction  by  4. 

48       48  -r-  16       3       We  thus  obtain  the  fraction  -f ,  whose  terms 
have  no  common  divisor.    The  fraction  is 
therefore  in  its  smallest  terms.     Or, 

Since  fractions  are  in  their  smallest  terms  when  their  numerator 
and  denominator  have  no  common  divisor,  to  reduce  them  to  their 
smallest  terms  we  may  divide  both  terms  by  their  greatest  common 
divisor. 

RULE. — Divide  the  numerator  and  denominator  by  any  common 
divisor,  and  continue  to  divide  thus  until  the  terms  have  no  common 
divisor,  Or, 

Divide  both  terms  by  their  greatest  common  divisor. 

2.  Reduce  ^f ,   $$,  |-f ,  $--0-,  to  their  smallest  terms. 

3.  Reduce  f|,  ffo,  |£,  •}-££,  to  their  smallest  terms. 


100 


COMMON    FRACTIONS. 


Keduce  to  their  smallest,  or  lowest  terms: 


4  n 

6.  -ift. 

7. 


9.  m- 

10.  B$. 

11.  &±* 

12. 
13. 


14. 
15. 


17. 

18.  ffff. 


19. 

20. 

21       1710 

22.    ^ffv 
23. 


CASE  III. 

163*  To    reduce    integers    or    mixed     numbers    to 
improper  fractions. 

1.  How  many  halves  are  there  in  1  apple?     In  4  apples? 
In  6  apples? 

2.  How  many  thirds  are  there  in  1  orange?     In  3  oranges? 
In  5  oranges? 

3.  How  many  fourths  are  there  in  2?     In  3?     In  4? 

4.  How  many  fifths  are  there  in  3?     In  4?     In  6? 

5.  How  many  fourths  are  there  in  1^?     In  2|? 

6.  How  many  thirds  are  there  in  2|?     In  3J  ?     In  6|? 


WRITTEN    EXERCISES. 

164.  1.  Reduce  8f  to  sevenths. 

PROCESS.  ANALYSIS. — Since   in    1    there     are   7 

g  __ .56.  sevenths,    in    8    there     are     8     times    7 

sevenths,  or  -576-;   and   in  8  +  f-   there   are 


RULE. — Multiply  the  integers  by  the  given  denominator,  to 
this  product  add  the  numerator  of  the  fractional  part,  if  there  be 
any,  and  write  the  result  over  the  given  denominator. 


2.  Change  5^  to  fourths. 

3.  Change  15  to  fifths. 


4.  Reduce  13^-  to  sixths. 

5.  Reduce  18^  to  elevenths. 


REDUCTION. 


6.  Change  5^-  to  ninths.     To  eighteenths. 

7.  Change  6^-  to  twelfths.     To  twenty-fourths. 

8.  Change  8T3¥  to  fourteenths.     To  forty-seconds. 

9.  Eeduce  9^  to  twentieths.     To  sixtieths. 

Reduce  to  improper  fractions: 


10.  13f. 

14.  25f 

18.  421ff. 

22.  867^- 

11.  12f 

15.  29-fr. 

19.  540i|. 

23.  904Jf. 

12.  18&. 

16.  37||. 

20.  763£f 

24.  314^. 

13.  23f. 

17.  56ff 

21.  419jfr. 

OF:     791  1  2 

^U.     i  ^/J.-T-2^"» 

CASE  IV. 

165.  To   reduce  improper   fractions   to   integers  or 
mixed  numbers. 

1.  How  many  days  are  there  in  6  half-days?     In  8  half- 
days?     In  14  half-days? 

2.  How  many  yards  are  there  in  9  thirds  of  a  yard  ?     In 
15  thirds?     In  18  thirds? 

3.  If  a  boy  pick  1  bushel  of  peaches  per  hour,  how  many 
bushels  can  he  pick  in  10  hours?   How  many  are  10  halves  ? 

4.  If  a  man  can  earn  ^  of  a  dollar  per  hour,  how  much 
can  he  earn  in  12  hours?     How  many  are  12  fourths? 

5.  How  many  units  are  there  in 

6.  How  many  dollars  are  there  in 


WRITTEN    EXERCISES. 


166.    1.  Reduce 


to  a  mixed  number. 


ANALYSIS.  —  Since  7  sevenths  equal 
1  unit,  123  sevenths  are  equal  to  as 
many  units  as  7  sevenths  are  contained 
,  or  17f  times.     Therefore  if3.  =  17f 


PROCESS. 
12_3  __  123  _^_  7  —  174 


times  in 

RULE.  —  Divide  the  numerator  by  the  denominator. 


105 


N    FRACTIONS. 


2.  Change  S-1^/  to  dollars. 

3.  Change  ±f£  pounds  to  pounds. 

4.  Change  -^-  ounces  to  ounces. 

5.  Change  if-5-  to  a  mixed  number. 

6.  Change  ^-  to  a  mixed  number. 

7.  Change  -^  to  a  mixed  number. 

Keduce  to  integers  or  mixed  numbers  : 


8. 

9. 
10. 
11. 


12. 
13. 
14. 
15. 


TO- 


16. 
17. 
18. 
19. 


20. 
21. 
22. 
23-  Wo4/- 


CASE  V. 

167.  To  reduce  dissimilar  fractions  to  similar  frac- 
tions. 

1.  How  many  fourths  are  there  in  \  of  an  orange? 

2.  How  many  sixths  of  a  field  are  there  in  ^  of  a  field? 

3.  How  many  eighths  in  f  ?     How  many  ninths  in  f  ? 

4.  Express  each  of  the  fractions  f ,  f ,  and  f  as  twelfths. 

5.  Express  each  of  the  fractions  ^  and  f  as  twentieths. 

6.  If  ^  is  divided  into  3  equal  parts,  how  large  is  each  part? 

7.  If  i  is  divided  into  2  equal  parts,  how  large  is  each  part? 

8.  When  \  and  \  are  divided  into  equal  parts,  what  parts 
are  common  to  both  ? 

9.  When  \  and  \  are  divided  into  equal  parts,  what  parts 
are  common  to  both  ? 

10.  What  equal  parts  are  common  to  both  \  and  -|-? 

11.  When  i,  ^  and  ^  are  divided  into  equal  parts,  what 
parts  are  common  to  all?  / 

12.  Change  ±  ,  1,  1,  to  equivalent  fractions  having  the  same 
fractional  unit.     Express  the  resulting  fractions  in  equivalent 
fractions  having  their  least  common  denominator. 


REDUCTION.  103 

Keduce  to  fractious  having  the  same  fractional  unit : 


11.  |  and  f. 

14.  f  and  T\. 

17.  -^ 

and 

12.  |  and  TV 

15.  •§•  and  -A?. 

O                           \.  A 

18.     y\ 

and 

13.  |  and  f. 

o                     t> 

16.  f  and  ^. 

19.  T52 

and 

168.  Similar  Fractions  are  those  that  have  the  same 
fractional  unit. 

169.  Dissimilar  Fractions  are  those  that  have  not 
the  same  fractional  unit. 

170.  Similar  fractions  have  a  Common   Denomi- 
nator. 

171.  When  similar  fractions  are  expressed  in  their  small- 
est terms  they  have  their  Least  Common  Denomi- 
nator. 

172.  PRINCIPLES. — 1.  A  common  denominator  of  two  or  more 
fractions  is  a  common  multiple  of  their  denominators. 

2.   The  least  common  denominator  of  tivo  or  more  fractions  is 
the  least  common  multiple  of  their  denominators. 


WRITTEN    EXERCISES. 

173.    1.  Reduce  f  and  -|  to  similar  fractions. 

PROCESS.  ANALYSIS. — Since  similar  fractions  have  a 

3.  _ -  3.  x  g  — - 14.         common  denominator,  to  make  these  fractions 
similar  we  must  change  them   to  equivalent 
IT  —  ¥><4~32         fractions  having  a  common  denominator. 

Since   a   common   denominator  of   two  or 

more  fractions  is  a  common  multiple  of  their  denominators  (Prin.), 
we  find  a  common  multiple  of  the  denominators  4  and  8,  which 
is  32. 

We  then  multiply  the  terms  of  each  fraction  by  such  a  number  as 
will  change  the  fraction  to  thirty-seconds. 


104  COMMON    FE ACTIONS. 

2.  Reduce  f ,  f  and  -f  to  similar  fractions  having  their 
least  common  denominator. 

PROCESS.  ANALYSIS. — The  least  common  denomina- 

£  __  2.  x  4  __  ^g,         tor  of   several  fractions  is  the  least  common 

multiple  of    their    denominators     (Prin.  2); 

1"  —  ¥  x  3  =  T2          therefore  we  find  the  least  common  multiple 
I  =  £  *  2  _  ii         of  3,  4,  and  6,  which  is  12. 

We  then  multiply  the  terms  of  each  frac- 
tion by  such  a  number  as  will  change  it  to  twelfths,  or  to  a  fraction 
whose  denominator  is  12.  Or, 

Since  1  is  equal  to  £f ,  |  is  equal  to  J  of  ^f ,  or  T4^,  and  f  are  equal 
to  2  times  T42,  or  T8^,  etc. 

RULE. — Find  the  common,  or  least  common  multiple  of  the 
denominators  for  a  common,  or  least  common  denominator. 

Divide  this  denominator  by  the  denominator  of  each  fraction 
and  multiply  both  terms  of  the  fraction  by  the  quotient 

Keduce  all  mixed  numbers  to  improper  fractions  and  all  fractions 
to  their  smallest  terms. 

Change  the  following  to  similar  fractions  having  their  least 
common  denominator : 


3. 


®*     5»    20'   "3TP 
"•    T>   T8>   "35"' 


7.  f  ,    f  ,  A- 

8.  |,  A,  A- 

"•      "5"'      2~0>    2T* 

10.  A.  if.  If  • 


11.  3f,  If,  #. 

19      13      14      36 

JLZ;.     o  5  j        5>    90* 

13.  H,  if.  if 

14.  M ,  A.  «• 


ADDITION. 

1.  James  has  2  fifths  of  a  dollar,  and  his  brother 
has  4  fifths  of  a  dollar.     How  many  fifths  have  both  ? 

2.  George  spent  $f  on  Monday,  and  $f  on  Tuesday.  How 
much  did  he  spend  in  both  days  ?  How  many  sevenths  are  f 
and  ? 


ADDITION.  105 

3.  Mr.  A.  sold  \  of  his  farm  at  one  time,  and  f  at  another 
time.     What  part  of  it  did  he  sell  ?     What  is  the  sum  of  -J- 
and  |? 

4.  James  caught  a  fish  in  the  morning  that  weighed  f  of  a 
pound,  and  another  in  the  afternoon  that  weighed  f  of  a  pound. 
What  did  both  weigh?     What  is  the  sum  of  f  and  f  ? 

5.  Marian   gave  $J  for  a  book  and  $^  for  some  wrriting 
paper.     How  much  did  she  pay  for  both?     What  is  the  sum 
of  £  and  i? 

6.  Ella  gave  f  of  her  apple   to  a  poor  beggar  and  Julia 
gave  him  \  of  hers.     How  many   fourths   did   he   receive? 
What  is  the  sum  of  f  and  \  ? 

7.  I  bought  \  of  an  acre  of  ground  for  a  site  for  a  house, 
and  \  of  an  acre  for  a  site  for  a  barn.     How  much  land  did  I 
buy?     What  is  the  sum  of  %  and  £?     Of  \  and  J?     Of  f 


8.  Mr.  A.  gave  $^-  to  one  man  and  $-|  to  another.     How 
much  did  he  give  to  both? 

9.  A  merchant  sold  f  of  a  bushel  of  clover  seed  to  one 
farmer,  and  f  of  a  bushel  to  another.     How  much  did  he 
sell  to  both  ? 

10.  Sarah  paid  $f  for  eggs  and  $-J  for  butter.     How  much 
did  both  cost  her  ? 

11.  I  paid  $f  for  turnips  and  $f  for  squashes.     How  much 
did  I  pay  for  both  ? 

12.  A  merchant  sold  f  of  a  yard  of  silk  to  one  lady  and  -f 
of  a  yard  to  another.     How  much  did  he  sell  to  both? 

13.  A  boy  earned  $-|  in  the  forenoon  and  8-g-  in  the  after- 
noon.    How  much  did  he  earn  that  day  ? 

14.  What  is  the  sum  of  f  and  Ty    £  and  T7^  ?    •&  and  T8¥  ? 

15.  What  kind  of  fractions  can  be  added  without  changing 
their  form? 

16.  What  must  be  done  to  dissimilar  fractions  before  they 
can  be  added?     How  are  dissimilar  fractions  made  similar? 


106  COMMON    FRACTIONS. 

175.  PRINCIPLES. — 1.   Only  similar  fractions  can  be  added. 
2.  Dissimilar  fractions  must  be  reduced  to  similar  fractions 

before  adding. 

WRITTEN     EXERCISES. 

176.  1.  What  is  the  sum  off,  f  and  f  ? 

PROCESS.  ANALYSIS. — Since  the  frac- 

I  +  f  +  |  =  f  £  +  H  +  A  =  M     ti0nS   ^  "^  Simllar'  bef°re 

adding  we  must  change  them 

to  similar  fractions,  or  equivalent  fractions  having  a  common  denom- 
inator. 

The  least  common  denominator  of  the  given  fractions  is  36;  and 
|  — 1^  1  =  11,  and  %  =  -£$.  Hence  the  sum  of  the  given  fractions 
must  be  equal  to  the  sum  of  f§,  f-J,  and  -f$,  which  is  f  f,  or  Iff. 

2.  What  is  the  sum  of  5},  6|  and  2f  ? 

PROCESS.  ANALYSIS. — Since  the  numbers  are  composed  of 

51  —  5  6  both  integers  and  fractions,  we  may  add  each  sep- 

£2 „  g  arately  and  unite  the  sums.     Thus,  the  sum  of  the 

1 2  fractional  parts  is  f  f ,  or  \\\  ;  the  sum  of  the  inte- 

2|  =  2^-  gers  is  13;  and  the  sum  of  both,  14  JJ. 

RULE. — Reduce  the  given  fractions  to  similar  fractions,  add 
their  numerators  and  write  the  sum  over  the  common  denomi- 
nator. 

When  there  are  mixed  numbers,  or  integers,  add  the  fractions 
and  integers  separately  and  then  add  the  results. 

If  the  sum  be  an  improper  fraction,  reduce  to  an  integral  or  mixed 
number. 


Find  the  sum 

3.  Of  f,  f ,  |,  |  and 

4.  Of  i  -I,  \,  i  and 

5.  Of  |,  \ ,  |,  |  and 

6.  Of  f ,  |,  f  $  and 


7.  Of  f ,  H,  |f  and  fi-. 

8.  Of  24-,  4,  3^  and  5f . 

9.  Of  27TV  8f  and  40f . 
10.  Of  13£,  15f  and  20|i. 


SUBTRACTION.  107 

Add  the  following: 

11       4.     15.     18.     _5          7 

7'    21'    14'    28'    35* 

12.  4f ,  5|,  8£,  2f,  7|,  4f. 

13.  9t,  7|,  8| 


14.  7T%,  8, 

15-  ft,  tf ,  2ft,  3A, 

16.  3f ,  4&,  6, 


17.  A  farmer  received  $18f  for  hay,  $65f  for  a  cow,  and 
$161f  for  a  horse.     How  much  did  he  receive  for  all? 

18.  A  man  earns  $67f  per  month,  and  each  of  his  two 
sons  $23f  per  month.     How  much  do  all  earn  per  month? 

19.  A  pedestrian  walked  45|  miles  on  Monday,  47f  on 
Tuesday,  50f  on  Wednesday.     How  far  did  he  walk? 

20.  A  has  5J  acres  of  land,  B  has  lOf  acres  more  than  A, 
C  has  as  much  as  both  A  and  B.     How  many  acres  have  B 
and  C  together? 

SUBTRACTION. 

177.    1.  Mary  earned   5  ninths  of  a  dollar  and  spent  2 
ninths.     How  many  ninths  of  a  dollar  had  she  left? 

2.  Mr.  A.  owning  ^  of  a  flouring  mill,  sold  |-  of  it.     How 
many  sevenths  did  he  then  own? 

3.  From  -f   subtract  f .     From  f  subtract  -|.     From  -^ 
subtract  -f^. 

4.  From  \\  subtract  -f%.     From  T8-g-  subtract  -^. 

5.  Mr.  B.  owned  a  lot  containing  |-  of  an  acre.     How  much 
had  he  left  after  selling  \  of  an  acre? 

6.  A  boy  paid  •$-§•  for  a  whip,  but  sold  it  after  a  time  for 
%\.     How  much  did  he  lose? 

7.  Find  the  difference  between  \  and  \.     \  and  -|. 

8.  What   kind    of   fractions    can    be    subtracted    without 
changing  their  form? 

9.  What  must  be  done  to  dissimilar  fractions  before  they 
can  be  subtracted?    How  are  dissimilar  fractions  made  similar? 


108  COMMON    FRACTIONS. 

178.  PRINCIPLES. — 1.  Only  similar  fractions  can  be  subtracted. 
2.  Dissimilar  fractions  must  be  reduced  to  similar  fractions 

before  subtracting. 

WRITTEN     EXERCISES. 

179.  1.  What  is  the  difference  between  ±±  and  -f  ? 

PROCESS.  ANALYSIS. — Since  the  fractions  are  not 

1 1 & 33 8__        similar,  before  subtracting  we  must  change 

them  to  similar  fractions. 

~3lT~  —  3~6  The  common  denominator  of  the  given 

fractions  is  36;  and  ij  =  f},  and  J  =  -^-. 

Hence  the  difference  between  the  given  fractions  is  equal  to  the  dif- 
ference between  f  f  and  ^8B-,  which  is  f  f . 

2.  What  is  the  difference  between  23J  and  4f . 

PROCESS.  ANALYSIS. — Since  the  numbers  are 

2  3  JL  — -  2  3  3  _  2  2  J-§-  composed  of  both  integers  and  frac- 
tions, we  may  subtract  each  sepa- 
rately. 

We  first  reduce  the  given  fractions 
to  similar  fractions.     Since  we  can 

not  take  |f  from  T3f,  we  unite  with  the  T32  1,  or  ||,  taken  from  23, 
making  -J-f .     Then  22^f  —  4}|  =  18j52,  the  remainder. 

RULE. — Reduce  the  fractions  to  similar  fractions. 

Find  the  difference  of  the  numerators  and  write  it  over  the  com- 
mon denominator. 

When  there  are  mixed  numbers  or  integers,  subtract  the  frac- 
tions and  integers  separately. 

Mixed  numbers  may  be  reduced  to  improper  fractions  and  sub- 
tracted according  to  the  first  part  of  the  rule. 

(3.)       (4.)        (5.)        (6.)        (7.)        (8.) 
From    |-  -§-  T63"  M  io"  ir 

Take     f  ^  TV  A  «  A 


MULTIPLICATION.  1 09 

9.  From   f  take  rV     |     15.  From  10££  take  ff. 


10.  From   f  take   f. 

11.  From  T67  take  T3-g-. 

12.  From  /¥  take  T\. 

13.  From  £$  take  ^. 

14.  From  ££  take  |f 


16.  From    66f  take  331 

17.  From.  210^  take  109f 

18.  From  112    take  75£. 

19.  From  606|  take  70J-. 

20.  From  589|  take  67f . 


21.  If  from  a  bin  containing  506|  tons  of  coal,  418|-  tons 
are  taken,  how  many  tons  still  remain? 

22.  A  lady  having  $25,  paid  $2|  for  a  pair  of  gloves,  $15f 
for  a  bonnet,  and  $3f  for  some  lace.     How  much  money  had 
she  left? 

23.  A  man  owned  a  farm  of  412  acres.     He  sold  three 
parcels  of  land  from  it,  the  first  containing  60f  acres,  the 
second  45^  acres,  and  the  third  116^-  acres.     How  many  acres 
did  he  sell,  and  how  many  had  he  remaining? 

24.  A  clerk  earned  $50|  per  month.     He  paid  $20f  for 
board,  $5f  for  washing,  and  $4f  for  other  expenses.     How 
much  did  he  save  per  month? 


MULTIPLICATION. 

CASE  I. 
180.  To  multiply  a  fraction  by  an  integer. 

1.  At  $-£•  a  yard  what  will  3  yards  of  cambric  cost? 

2.  If  a  man  can  earn  $^  per  hour,  how  much  can  he  earn 
in  5  hours?     How  much  can  he  earn  in  8  hours? 

3.  James  gave  f  of  an  apple  to  each  of  5  children.     How 
many  apples  did  he  give  to  all?     How  mtfch  is  5  times  f? 

4.  How  many  fifths  are  there  in  6  times  £?     In  7  times  f  ? 

5.  If  Mr.  A.  spends  $2^-  per  day,  how  much  will  he  spend 
in  5  days?     How  much  in  10  days? 


110 


COMMON    FRACTIONS. 


6.  How  much  is  2   times  -f-  ?     How  does  the  result  com- 
pare with  f  ?     How  is  it  obtained  from  |? 

7.  In  multiplying  a  fraction  what  part  of  the  fraction  do 
we  multiply? 

8.  Multiply  f  by  2.     f  by  3.     -£-  by  7. 

9.  Express  2  times   f  in  smallest  terms.     How    is   this 
result  obtained  from  the  fraction  f  ? 

10.  In  what  other  way  then  may  we  multiply  a  fraction? 

11.  How  much  is  3  times  f  ?     4  times  f  ?     6  times  -f^. 

12.  How  much  is  5  times  -|?     6  times  -f  ?     9  times  f  ? 

13.  How  much  is  4  times  -|?     3  times  T5^?     5  times  -J  ? 

14.  How  much  is  4  times  f  ?     7  times  ^-?     9  times 


181.  PRINCIPLE.  —  Multiplying  the  numerator  or  dividing  the 
denominator  of  a  fraction  by  any  number,  multiplies  the  fraction 
by  that  number. 


WRITTEN    EXERCISES. 


1.  Multiply  |f  by  6. 

PROCESS. 


Or, 


ANALYSIS. — 6  times  13  twen- 
ty-fourths are  78  twenty -fourths, 
or  3£.  Or, 

Since  dividing  the  denomi- 
nator multiplies  the  fraction 
(Prin.),  6  times  Jf  are  ^,  or  3£. 


RULE. — Multiply  the  numerator  or  divide  the  denominator  by 
the  integer. 


Multiply  : 

Multiply  : 

Multiply: 

2.  A  by  5. 

7.  ||  by  7. 

12.  «  by  9. 

3.  A  by  7.    • 

8.  A  by  6. 

13.  if  by  13. 

4-  A  ^  5. 

9.  Tfr  by  8. 

14.  if  by  14. 

5.  &  by  3. 

10.  if  by  3. 

15.  if  by  18. 

6.  .tf  by  17. 

11.  -Lf  by  7. 

16.  M  by  75. 

MULTIPLICATION.  Ill 

17.  What  is  the  value  of  a  load  of  17  bushels  of  apples  at 
$1  a  bushel? 

18.  If  a  boy  earns  $f  per  day,  how  much  can  he  earn  in 
9  days? 

19.  At  87|  a  barrel  what  will  7  barrels  of  flour  cost? 

PROCESS.          ANALYSIS. — In  multiplying  a  mixed  number,  we  mul- 

$  7  f-  tiply  the  fractional  part  and  integer  separately  and  add 

7  the  results. 

~~^~I  Thus,  7  times  $£  =  $y  =  $5  J.    7  times  $7  are  $49,  and 

4  9  4  the  sum  of  $49  and  $5}  is  $54J. 

We  may  reduce  the  mixed  number  to   an  improper 

•*  fraction  before  multiplying. 

20.  If  a  man  travel  21f  miles  per  day,  how  far  can  he- 
travel  in  4  days? 

21.  What  will  13  yards  of  cloth  cost  at  $6|  a  yard? 

22.  If  a  steamship  sails  17^-  miles  an  hour,  how  far  can 
she  sail  in  9  hours? 

CASE  II. 

182.  To  multiply  an  integer  by  a  fraction. 

1.  Henry  had   6  rabbits  and  sold  ^  of  them  to  James. 
How  many  did  he  sell? 

2.  Jane  had  16  cherries  and  gave  \  of  them  to  her  sister. 
How  many  did  she  give  to  her  sister? 

3.  How  much  is  £  of  $18?    f  of  $18? 

4.  How  much  is  |  of  7  apples?    1  of  9  bushels?    £  of  5 
ounces?     \  of  3  lemons? 

5.  How  much  is  £  of  5?    f  of  5?    f  of  5? 

6.  How  much  is  f  of  7?    f  of  8?    f  of  12? 

7.  What  is  f  of  36?    f  of  32?    £  of  54? 

8.  How  much  is  f  of  35  tons  ?    f-  of  49  horses  ?    f  of  80? 

183.  PRINCIPLE. — Multiplying  by  a  fraction  is  taking  such  a 
part  of  a  number  as  is  imlicated  by  the  fraction. 


112  COMMON    FRACTIONS. 


WRITTEN     EXEJtCI  SES. 

1.  Multiply  75  by  f? 

PROCESS.  ANALYSIS.  —  To  multiply  75  by  f  is 

1  5  to  find  f  of  75.     |  =  3  times  \.    ±  of 

7  5  X  f  —  fr'*$x3'  =  45        75  is  15>  and  f  =  3  times  15,  or  45. 

Or, 
Since  f  =  £  of  3,  f  of  75  =  £  of  3  times  75,  or  -*Ap  =  45. 


RULE.  —  Multiply  the  integer  by  the  numerator  of  the  multiplier, 
and  divide  the  product  by  the  denominator. 
When  possible  use  cancellation. 


2.  Multiply      9  by  TV 

3.  Multiply    17  by  &. 

4.  Multiply    12  by  £f. 

5.  Multiply    18  by  ff. 

6.  Multiply  100  by  ^. 

7.  Multiply  144  by  ft. 


8.  Multiply      51  by  T5T. 

9.  Multiply      79  by  T<V. 

10.  Multiply  8318  by  &. 

11.  Multiply  $406  by  T\. 

12.  Multiply  $718  by  |f 

13.  Multiply  $825  by  ff . 


14.  A  man  owned  a  mill  worth  $7850.     How  much  money 
should  he  receive  for  f  of  it? 

15.  A  sportsman  shot   48    birds  one  day,  and  f  as  many 
the  next.     How  many  did  he  shoot  in  both  days? 

16.  Multiply  46  by  5f . 

PROCESS.  ANA  LYSIS. — In    m  u  Iti  ply  i  ng 

4g        Qr  53.  _ _  2_8  by  a  mixed  number  we  inulti" 

53.  4fiv28 1288      ply  ky  tne  integer  and  fraction 

<T^r  1  0  858 o  1 7  ,s     separately  and  add  the  products. 

*  <t  -^5—  -  ^  °  <  %    Thus,  f  of  46  is  ijU,  or  27$.     5 

times  46  =  230,  and  27|  +  230 

257|  =257$,  the   product.      Or,   we 

may  reduce  the  mixed  number 
to  an  improper  fraction  and  multiply. 

17.  Multiply  36  by  5i;     6f;     7f; 


MULTIPLICATION  113 

CASE   III. 
184.  To  multiply  a  Traction  by  a  fraction. 

1.  How  much  is  \  of  4  fifths  of  a  yard?     How  much  is  \ 
of  3  ninths  of  a  yard?     \  of  8  twelfths  of  a  yard? 

2.  If  James  has  f  of  a  dollar,  and  Anna  has  \  as  much, 
how  much  has  Anna?     If  Henry  has  -|  as  much  as  James, 
how  much  has  he?     How  much  is  \  of  -I?     \  of  I- ? 

o  o  o  o 

3.  A  man  who  owned  f  of  a  steamship  sold  \  of  his  share. 
What  part  of  the  vessel  did  he  sell?     How  much  is  \  of  f  ? 

4.  How  much  is  \  of  ||?    f  of  |f?     f  of  ff  ?     -J  of  T8g  ? 

5.  If  |  of  a  yard  be  divided  into  2  equal  parts,  what  part 
of  a  yard  will  each  part  be?     How  much  is  \  of  \  yard?    \ 
of  ^  yard? 

6.  At  %\  a  bushel  what  will  \  of  a  bushel  of  oats  cost? 
How  much  is  \  of  i?     iof|?     \  of  i?     fofi?     fof|? 

7.  What  is  the  value  of  \  of  |?    £  of  f?    f  of  J?    f  of  f  ? 

WRITTEN    EXERCISES. 

1.  Multiply  f  by  f . 

PROCESS.  ANALYSIS. — To  multiply  f  by  f  is  to 

4V   3_4X3_1_2        find  t  of  Y>   or  3  times  i  Of    f        i  of    7  = 

7Jy,  and  f  are  3  times  ^  or  ^  =  H- 

RULE. — Reduce  all  integers  and  mixed  numbers  to  improper 
fractions. 

Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  together  for  its  denominator. 

1.  When  possible  use  cancellation. 

2.  The  word  of  between  fractions  is  equivalent  to  the  sign  of  multipli- 
cation.   Such  expressions  are  sometimes  called  compound  fractions.  Thus, 
J  of  |  is  equal  to  J  X  £• 

3.  Integers  may  be  expressed  in  the  form  of  fractions  by  writing  1 
as  a  denominator.     Thus,  4  may  be  written  as  J. 

8 


114 


COMMON    FKACTIOJSS. 


Multiply  : 

2.  -h  by  f 

3.    |  byf 

4»  ••  ^   bv  —  . 

Multiply  : 

5.    |  by  f 
6.  TV  by  ^. 

7.  Jt  by  |. 

Multiply: 


9- 


10.  Mby*. 


Find  the  value  of 

11.      xx 


14. 

is. 

16. 


17.    ||  X  U  X 

18. 


19. 
20. 
21. 
22. 


I    X 

I  x 
f  x 


X 

x 

x 


X  f 

xT58- 
x  if 


ft 


23.  Multiply  f  of  1|  by  f  o 

PROCESS. 
f  X  |  X  |  X  T92-  X  f  =  4  ,  or  14 


of  4. 


ANALYSIS.  —  All  the  mixed 
Mmbera  must  be  changed  to 
improper  fractions,  and  all 

whole  numbers  expressed  in  the  fractional  form.     Multiplying  and 

cancelling  we  have  f,  or  1  J. 


24.  Multiply  |  of  f  of  5  by  -^  of  |  of  3£. 

25.  Multiply  f  of  -^  of  8  by  f  of  T9<>  of  15. 

26.  Multiply  3£  times  f  by  4  times  f  of  7. 

27.  Multiply  5£  times  £  of  18  by  f  of  3  times  4-  of  4. 

28.  What  will  be  the  cost  of  T9¥  of  a  yard  of  cloth  at  $f  a 
yard? 

29.  16^-  feet  make  a  rod.     How  many  feet  are  there  in  5| 
rods? 

30.  A  man  who  Owned  f  of  a  mill,  sold  -f-  of  his  share. 
What  part.  of  the  mill  did  he  sell? 

31.  How  many  yards  of  cloth  are  there  in  12^  pieces  of 
cloth,  each  piece  containing  42f  yards? 

32.  At  $16f  per  ton  how  much  can  be  realized  from  the 
sale  of  4f  tons  of  hay  ? 


DIVISION.  115 


33.  What  is  the  value  of  f  of  -fr  of  |£  of  -^  of  15? 

34.  What  is  the  value  of  ^  of  3f-  of  ff  of  29? 


DIVISION. 

CASE  I. 
185.  To  divide  a  fraction  by  an  integer. 

1.  Mr.  Allen  divided  3  fourths  of  a  dollar  equally  between 
3  boys.     How  much  did  each  receive? 

2.  A  man  divided  f  of  an  acre  into  3  equal  lots.     How 
large  were  they? 

3.  If  7  yards  of  cloth  were  bought  for  $$,  what  was  the 
-cost  per  yard? 

4.  If  4  books  cost  $if ,  what  is  the  cost  of  each? 

5.  Mr.  Hurd  put  f  of  his  crop  of  wheat  on  3  wagons. 
What  part  of  his  crop  was  on  each  wagon?      How  much 
isf-3?  _ 

6.  In  dividing  a  fraction,  what  part  of  the  fraction  is 
divided? 

7.  A  gentleman  diyided  ^  a  barrel  of  flour  equally  between 
2  people.     What  part  of  a  barrel  did  each  receive?     How 
much  is  \  -r-  2  ? 

8.  3  boys  spent  altogether  $J.     If  each  spent  the  same 
amount,  what  part  of  a  dollar  did  each  spend?     How  much 
is  i-4-3? 

9.  Mr,  Smith  divided  ^  of  his  farm  into  3  equal  fields. 
What  part  of  his  farm  did  each  field  contain  ?     How  much 
is  -J-  -=-  3?     How  is  the  result  obtained  from  the  fraction  ^? 

10.  In  what  other  way,  then,  besides  dividing  the  numera- 
tor may  a  fraction  be  divided  ? 

11.  When  a  number  is  divided  by  3  what  part  of  it  is  found? 

12.  When  a  fraction  is  divided  by  7  what  part  of  it  is  found? 


116 


COMMON    FRACTIONS. 


13.  What  is  1  off?    f--3?    i 

14.  What  is  the  value  of  £ -r-  2?     Of  f  •+-  3?     Of  -£- ~-  9? 

186.  PRINCIPLE. — Dividing  the  numerator  or  multiplying  the 
denominator  of  a  fraction  by  any  number,  divides  the  fraction  by 
that  number. 


WRITTEN    EXERCISES. 


1.  Divide  ff-  by  6. 

PROCESS.  ANALYSIS. — Since  dividing  the  nu- 

merator of  a  fraction  divides  the  frac- 
tion, the  fraction  |f  may  be  divided  by 
6  by  dividing  the  numerator  by  6.  Or, 
Since  multiplying  the  denominator 
divides  the  fraction,  the  fraction  may 
be  divided  by  6  by  multiplying  the  denominator  by  6.  The  result  by 
both  processes  is  T27. 

RULE. — Divide  the  numerator  or  multiply  the  denominator  by 
the  given  integer. 


Or, 

TT  ~^~  ®==  TTx  6" I=r  TT 


Divide  : 

Divide  : 

Divide  : 

2-    f 

by  4. 

6    ?4 

by  8. 

10.  |f 

by  21. 

3.    | 

by  8. 

7.  45 

by  15. 

11.  ft 

by  18. 

4    12 

2  9 

by  3. 

8-  |I 

by  7. 

12.  1! 

by  38. 

5.  || 

by  6. 

981 
•    T3 

by  18. 

13.  It 

by  35. 

14.  Divide  16f  by  5. 


PROCESS. 


Or,     5)16f 


ANALYSIS. — We  may  reduce  16|  to 
an  improper  fraction,  and  divide  as 
before.  Or, 

We  may  divide  without  reducing  to 
an  improper  fraction.  Thus,  5  is  con- 
tained in  16 1,  3  times  and  a  remainder 
of  If,  or  |,  and  J  divided  by  5,  equals 
-fa.  Therefore  the  quotient  is  32V 


DIVISION.  117 


Divide : 

15.  17f  by  6. 

16.  25f  by  4. 


Divide : 

17.  38f    by  8. 

18.  24fV  by  9. 


Divide : 

19.  361  by  10. 

20.  25f  by  7. 


21.  A  man  gave  each  of  his  5  sons  an  equal  share  of  -f 
of  his  estate.     What  part  of  the  whole  did  each  receive? 

22.  8  men  built  |  of  a  mile  of  wall  in  10  days.     What 
part  of  a  mile  did  each  man  build  daily? 

23.  Mr.  B.  earned  $35£  by  working  8  days.     How  much 
did  he  earn  per  day? 

24.  I  paid  $10|  for  27  pounds  of  butter.     What  did  I  pay 
a  pound? 

25.  A  farmer  realized  $233^  for  21  bushels  of  clover  seed. 
How  much  did  he  get  per  bushel? 

26.  Three  men  who  have  been  partners  in  business  gain 
$3216^.     If  they  share  equally,  what  will  be  each  one's  part 
of  the  gain  ? 

CASE  II. 
187.    To  divide  an  integer  by  a  fraction. 

1.  How  many  fourths  are  there  in  an  apple?     In  2  apples? 

2.  How  many  apples,  at  ^  cent  each,  can  be  bought  for  2 
cents?     For  3  cents?     For  10  cents? 

3.  At  $-|-  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $1?     For  $2?     For  $3?     For  $10? 

4.  At  $-J-  an  ounce,  how  many  ounces  of  nutmegs  can  be 
bought  for  $2?     How  many  at  $f  an  ounce  can  be  bought 
for  $2? 

5.  If  a  man  can  mow  -f  of  a  field  per  day,  how  long  will 
it  take  him  to  mow  the  entire  field?     How  long  if  he  can 
mow  |-  per  day? 

6.  At  %\  a  bushel,  how  many  bushels  of  lime  can  be  bought 
for  $5?     At  $f  a  bushel  how  many  bushels  can  be  bought 
for«5? 


118 


COMMON    FRACTIONS. 


7.  When  apples  are  worth  f  of  a  cent  apiece,  how  many 
can  be  bought  for  12  cents?     How  many  times  is  f  contained 
in  12? 

8.  How  many  pieces  of  cloth  f  of  a  yard  long  can  be  cut 
from  a  piece  9  yards  long?     How  many  times  is  f  contained 
in  9? 

9.  How  many  times  is  f  contained  in  8?     f  in  9? 

10.  What  is  the  quotient  when  8  is  divided  by  f?    10  by  f  ? 

11.  What  is  the  value  of  10--|?     7--|?     9---f? 


WRITTEN    EXERCISES. 


1.  Divide  12  by  f 


PROCESS. 


Or, 


ANALYSIS. — \  is  contained  in  12 
7  times  12,  or  84  times;  and  f,  one- 
sixth  of  84  times,  or  14  times.  Or, 

Reducing  12  to  sevenths,  we  have 
84  sevenths.  6  sevenths  are  contained 
in  84  sevenths  14  times. 


RULE. — Multiply  the  integer  by  the  denominator  of  the  fraction 
and  divide  the  product  by  the  numerator.  Or, 

Reduce  the  dividend  and  the  divisor  to  similar  fractions,  and 
divide  the  numerator  of  the  dividend  by  the  numerator  of  the 
divisor. 


Divide  : 

Divide  : 

Divide: 

2.  18  by  f. 

10.  72  by  f  . 

18.  51  by  if 

3.  64  by  f  . 

11.  34  by  f 

19.  69  by  |f. 

4.  26  by  \. 

12.  15  by  ft. 

20.  70by£f 

5.  48  by  f  . 

13.  49  by  fi 

21.  65  by  |f. 

6.  75  by  i-f  . 

14.  91  by  V3- 

22.  90  by  ff  . 

7.  31  by  A- 

15.  64  by  if 

23.  36  by  i-f. 

8.  45  by  |f 

16.  54  by  ff. 

24.  39  by  ff  . 

9.  39  by  TV 

17.  39  by  if 

25.  24  by  #. 

DIVISION.  119 

26.  Divide  24  by  3f 

PROCESS.  ANALYSIS. — When  the  divisor  is  a 

oj^ 2.  mixed  number  we  reduce  it  to  an  im- 

proper fraction  and  proceed  according 
=  6  f       to  the  rule.     Thus,  3J  =  J;  and  24  di- 
vided by  J  =  48,  and  by  }  =  \  of  48, 


Or,       31 

2_ 
7 


24  or  6f.    Or, 

We  may  reduce  3J-  to  halves,  and 
48  24  to  halves,  and  divide  the  numera- 

~7T^  tor  of  the  dividend  by  the  numerator 


of  the  divisor. 
Divide  the  following  by  both  processes : 


27.  15  by  3$. 

28.  23  by  6f. 


29.  26  by  7f . 

30.  35  by  6f 


31.  39  by  8f. 

32.  46  by 


33.  At  $f  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $15? 

34.  When  corn  is  worth  $£  a  bushel,  how  many  bushels 
must  a  man  sell  to  get  money  enough  to  pay  $18  taxes? 

35.  A  man  invested  $32  in  peaches  at  $lf  per  basket. 
How  many  baskets  did  he  purchase? 

36.  What  is  the  price  of  hay,  when  34  tons  sell  for  $37? 

37.  Mr.    Shaw   paid   $6  for   14|   pounds  of  Java  coffee. 
What  did  it  cost  him  per  pound? 

38.  It  requires  656  pounds  of  meat  to  supply  34  soldiers 
3-|  weeks.     How  much  does  each  soldier  eat  daily? 

CASE  III. 
188.  To  divide  a  fraction  by  a  fraction. 

1.  How  many  fourths  are  there  in  1?     How  many  fifths? 
eighths  ?     tenths  ?     fifteenths  ?     twentieths  ? 

2.  How  many  pieces  of  cloth  -i-  of  a  yard  long,  can  be 
cut  from  a  yard? 


120  COMMON    FRACTIONS. 

3.  If  the  pieces  were  J-  of  a  yard  long,  how  many  would 
there  be  ?     How  does  the  number  compare  with  the  number 
when  the  pieces  are  -|-  of  a  yard  long? 

4.  If  the  pieces  were  f  of  a  yard  long,  how  many  would 
there  be?     How  does  the  number  of  pieces  compare  with  the 
number  when  the  pieces  are  ^  of  a  yard  long? 

5.  How  many  times  is  ^  contained  inl?    f  ?    f  ?    |?    f? 

6.  Since  -g-  is  contained  in  1,  eight  times,  how  many  times 
will  it  be  contained  in  |?     What  part  of  8  times  will  it  be 
contained  in  ^-? 

7.  Since  f  is  contained  in  1  -^  of  8  times,  or  f  times,  how 
many  times  will  it  be  contained  in  ^?    How  many  times  in  ^? 


8.  What  is  the  value  of  1--|?    Of  1-f  ?    f 

9.  Into  how  many  parts  of  3  eighths  of  a  dollar  each,  can 
6  eighths  of  a  dollar  be  divided  ? 

10.  How  many  sacks  containing  ^  of  a  barrel  each,  can  be 
filled  from  -f$  of  a  barrel  of  flour?     How  many  times  is  ^ 
contained  in  T%?     f  in  f  ?     ^  in  ^  ?     ^  in  ^? 

11.  How  many  pine-apples  at  $J  each,  can  be  bought  for  $1  ? 

12.  How  many  times  is  ^  contained  in  ^?    ^  in  ^?    f  in  f  ? 

13.  How  many  times  is  ^  contained  in  1?     In  £?     In  ^? 

14.  How  many  times  is  |  contained  in  1?     In  i?     In  f  ? 

WRITTEN    EXERCISES. 

1.  Divide  ^  by  f. 

PROCESS.  ANALYSIS.  —  ^  is  contained  in  1,  5 

4  _._  &  _  ±  \/   5.—  20       times;   and  f  is  contained  in  1,  one- 

third  of  5  times,  or  f  times. 

Or,  And  since  f  is  contained   in  1,  f 

4_._£  _  2o_i_.2JL  _  2_o       times,  in  f  it  will  be  contained  f-  of 

^      f  =  if  times.     Or, 

\  is  equal  to  f§,  and  f  is  equal  to  f  J.     21  thirty-fifths  are  contained 
in  20  thirty-fifths  ff  times. 


DIVISION. 


121 


RULE. — Multiply  the  dividend  by  the  divisor  inverted.  Or, 
Reduce  the  dividend  and  divisor  to  similar  fractions  and  divide 
the  numerator  of  the  dividend  by  the  numerator  of  the  divisor. 

When  possible  use  cancellation. 


Divide: 

Divide  : 

Divide  : 

2.  k-  by  f 

3.  MbyM- 

4-  H  by  &. 

5-  If  by  A. 

6-  if  by  A. 

7.  if  by  f  . 
8.  ff  by  A- 

9-  M  by  A- 

10.  ifbyTV 
H-  Mbyif. 
12-  ft  by  A. 
13.  fi  by  &. 

14.  What  is  the  quotient  of  f  of  f  of  5|  divided  by  f  of  ^ 

off? 

PROCESS.  ANALYSIS. — In  the  so- 

|  of  f  of  -1/  _^_  I  of  -f-  of  |-  =  huion   of   examples  like 

0  this,  all  mixed  numbers 

i  X  4  X  "^  X  4  X  -^  X  -^  =  3-^  =  6-i    should  be  changed  to  im- 
proper fractions,  and  all 

fractions  that  are  factors  of  the  divisor,  inverted,  and  the  product  found 
as  in  previous  examples. 

15.  Divide  f  of  f  of  16  by  f  of  f  of  5J-. 

16.  Divide  f  of  ||  of  5J  by  4J  times  £  of  16. 

17.  Divide  £  of  f  of  f  by  f  of  f. 

18.  Divide  J-  of  f  of  T9T  by  5  times  f  of  f . 

19.  Divide  |  of  f  of  15  by  f  of  §  of  6. 

20.  Divide  f  of  T\  of  22  by  T\  of  f  of  16. 

21.  Divide  -&  of  3|  of  6  by  f  of  6  times  If. 

22.  Divide  8}-  times  i  of  7^  by  f  of  f  of  5. 

23.  How  many  pieces  of  ribbon  T2^  of  a  yard  in  length, 
can  be  made  from  ^  of  T9^  of  a  yard  ? 

24.  If  a  man  spends  $|  per  day  for  cigars,  in  how  many 
days  will  he  spend  $17£? 

25.  How  many  yards  of  cloth  at  $3f  per  yard  can  be 
bought  for  $317f  ? 


122  COMMON   FBACTIONS. 

26.  At  $f  per  bushel,  how  many  bushels  of  potatoes  can 
be  bought  for  61 7J? 

27.  If  a  family  uses  f  of  a  barrel  of  flour  a  week,  how  long 
will  5}  barrels  last? 

28.  If  a  boy  earns  $f  daily,  how  long  will  it  take  him  to 
earn  $3f  ? 

29.  A  certain    number   multiplied  by  f  is    equal*  to   f^-. 
AVhat  is  that  number? 

30.  If  a  man  can  saw  1^  cords  of  wood  in  one  day,  how 
long  will  he  require  to  saw  17^-  cords? 

31.  If  a  horse  eats  12^  bushels  of  oats  in  5  weeks,  how 
much  does  he  eat  in  a  day? 

32.  When  wheat  is  selling  at  81£  per  bushel,  how  many 
bushels  can  be  bought  for  $3168? 


FRACTIONAL  FORMS. 

189.  Expressions  of  unexecuted  division  are  often  written 
in  the  form  of  a  fraction. 

190.  A  fractional  form  having  an  integral  denominator 
and  a  fractional  numerator  is  called  a  Complex  Frac- 
tion. 

3  A 1 

Thus,  ~  and  ~  are  complex  fractions, 
o  y 

Expressions  which  have  a  fraction  in  the  "denominator  can  not 
properly  be  regarded  as  Complex  Fractions,  though  they  are  commonly 
classified  as  such. 

1.  Find  the  value  of  the  fractional  form  1C. 

PROCESS.  ANALYSIS. —  -?-   is    an    expression 

A  4  i 

JL  =  •§•  -=-  .£  =4-  X  -f~  -fy     °f  division,  and  is  the  same  as  f  -f- 1, 

"8  3  which  is  equal  to  |f. 


FRACTIONAL    RELATION    OF    NUMBERS. 


123 


Reduce  to  simple  fractions: 
6.  A. 


7.1. 


3, 

4. 

5. 


A 


9. 


5* 


10. 


11.  Li. 
A 

12.  H. 

¥ 


13. 


H 


14. 

16. 

17. 


|  of  9 
of  3 
T~ 
3 


FRACTIONAL  RELATION  OF  NUMBERS. 

CASE  I. 
191.  To  find  the  relation  of  one  number  to  another. 

1.  What  part  of  5  cents  is  1  cent?     2  cents?     3  cents? 
4  cents? 

2.  What  part  of  9  acres  is  5  acres?    7  acres?    3  acres?    4 
acres  ? 

3.  What  part  of  4  apples  is  1  apple?     ^  of  1  apple?     f 
of  1  apple?    |  of  1  apple? 

4.  What  part  of  $5  is  $2?     $|?    *i?     $i?     $f? 

5.  What  part  of  $6  is  $1?     $£?     $|?     $f  ? 

6.  Henry  had  $5  and  gave  his  brother  $f .     What  part  of 
his  money  did  he  give  his  brother? 

7.  James  earned  17,  and  his  brother  $2.     What  part  of 
the  whole  did  each  earn? 

PRINCIPLE. — Only  like  numbers  can  have  relation  to  each  other. 

8.  What  is  the  relation  of  5  to  9? 

ANALYSIS.— 1  is  $  of  9,  and  5  is  5  times  £  of  9,  or,  f . 
Hence  5  is  4  of  9. 


124  COMMON    FRACTIONS. 

What  is  the  relation 


9.  Of  7  to  21? 

10.  Of  12  to  16? 

11.  Of  10  to  28? 


12.  Of  9  to  18? 

13.  Of  8  to  32? 

14.  Of  32  to  48? 


15.  Of  15  to  24? 

16.  Of  14  to  35? 

17.  Of  18  to  81? 


18.  What  is  the  relation  of  4  to  2  ? 


ANALYSIS.—  1  is  J  of  2,  and  f  of  1  is  f  of  \  of  2,  or 
Hence  f  is  T%  of  2. 


of  2. 


What  is  the  relation 


19.  Of  |  to  4? 

20.  Of  %  to  9? 

21.  Of  |  to  6? 


22.  Of  |  to    8? 

23.  Of  |  to  15? 

24.  Of      to  25? 


25.  Of  |  to    6? 

26.  Of  f  to  18? 

27.  Of  |  to  12? 


28.  What  is  the  relation  of  f  to  -f  ? 

ANALYSIS. — \  is  \  of  f,  and  1  is  7  times  \  of  f,  or  J  of  f ;  and  since 
1  is  |  of  f,  |  of  1  is  f  of  |  of  f ,  or  Jf  of  f . 
Hence  f  is  ||  of  f- . 


What  is  the  relation 


29.  Of  |   to  |? 

30.  Of  f  to  |? 

31.  Of       to  4? 


32.  Of  f  to 

33.  Of  |  to   |  ? 

34.  Of  f  to   f  ? 


35.  Of 

36.  Of 

37.  Of 


to 
to 
to 


CASE   II. 

192.  A  number  and  its  relation  to  another  number 
given,  to  find  the  other  number. 

1.  2  cents  are  ^  of  how  many  cents? 
^  of  how  many  cents? 

2.  3  is  ^  of  what  number?     \  of  what  number? 
number? 

3.  8  is  ^  of  what  number?     -|  of  what  number?     |-  of  what 
number  ? 


of  how  many  cents? 
of  what 


REVIEW    EXERCISES. 


125 


4  12  is  -J-  of  what  number? 
what  number? 

5.  \  is  ^  of  what  number? 

6.  f  is  ^  of  what  number? 


f  of  what  number? 

of  what  number? 
of  what  number? 


f  of 


7.  24  is  f  of  what  number? 

ANALYSIS.  —  Since  24  is  f  of  a  certain  number,  1  fifth  of  the  num- 
ber is  J  of  24,  or  6;  and  since  6  is  ^  of  the  number,  the  number  must 
be  5  times  6,  or  30.  Hence  24  is  f  of  30. 


8. 


24  is  |  of  what  number  ? 


9.   18  is  f  of  what  number? 

10.  24  is  ^  of  what  number? 

11.  45  is  f  of  what  number? 


56  is  f-  of  what  number? 

49  is  -J  of  what  number? 

42  is  f  of  what  number  ? 

96  is  of  what  number? 


12.  ^  is  |  of  what  number? 

ANALYSIS.  —  Since  f  is  f  of  a  certain  number,  J  of  the  number  is  J 


of  f  ,  or  f  ;  and  since 
f  ,  or  f.     Hence  y  is 


of  the  number  is  f  ,  the  number  must  be  3  times 

of  f  . 


13.  -f^-  is  4  of  what  number?  -|f  is  - 

14.  J-f  is  f  of  what  number?  f|  is 

15.  ff  is  T73-  of  what  number  ?  f|  is 

16.  -   is   9-  of  what  number  ?          is 


of  what  number? 
of  what  number  ? 
of  what  number? 
of  what  number  ? 


REVIEW  EXERCISES. 

193.  1.  Mr.  B.  bought  a  barrel  of  flour  for  $7|,  a  cord  of 
wood  for  S5-|,  and  gave  the  clerk  a  twenty-dollar  bill.  How 
much  change  should  Mr.  B.  receive  ? 

2.  A  merchant  bought  360  pounds  of  sugar  at  11|  cents 
a  pound,  50  pounds  of  tea  at  621  cents  a  pound.    How  much 
did  he  pay  for  both  ? 

3.  If  a  man  can  cut  in  one  day  ^  of  a  field  containing  7 
acres  of  wheat,  how  many  acres  can  he  cut  in  ^  of  a  day  ? 


126  COMMON    FRACTIONS. 

4.  What  will  be  the  cost  of  3^-  dozen  eggs  at  18|  cents  a 
dozen  ? 

5.  If  a  man  can  hoe  a  field  in  7^  days,  how  long  will  it 
take  3  men  to  hoe  a  field  2-|  times  as  large  ? 

6.  From  a  barrel  of  kerosene  containing  41^  gallons  it  was 
estimated  that  -f  leaked  out.     If  I  paid  $6.15  for  it,  at  what 
price  per  gallon  must  I  sell  the  remainder  to  balance  the  loss 
sustained  by  leakage  ? 

7.  James  Henderson  sold  f  of  his  farm  of  155  acres  to 
Mr.  Paine,  and  Mr.  Paine  soon  sold  f  of  what  he  had  bought, 
to  Mr.  Banker.     How  many  acres  did  Mr.  Banker  buy  ? 

8.  Mr.  A.  built  a  block  of  stores  which  cost  him  $3122£ 
for  brick,  $1368^  for  lumber,.  $3258$  for  labor,  and  $1325^ 
for  other  expenses.     He  sold  the  block  for  $10000.     Did  he 
gain  or  lose,  and  how  much? 

9.  There  are  272^  square   feet  in  a  square  rod.     How 
many  square  feet  are  there  in  f  of  a  square  rod  ? 

10.  I  sold  a  house  and  lot  for  $3215,  which  was  If  times 
what  it  cost  me.     How  much  did  it  cost? 

11.  How  much  will  8  carpenters  earn  in  6|  days  at  $2-| 
per  day? 

12.  If  a  man  walks  3J  miles  per  hour,  in  how  many  hours 
can  he  walk  30^  miles? 

13.  Mr.  Jones  left  an  estate  valued  at   $19000.     f  of  it 
was  divided  equally  among  4  sons  and  the  rest  equally  among 
3  daughters.     What  was  the  share  of  each? 

14.  The  price  of  maple  sugar  this  year  is  only  -J  of  what  it 
was  last  year.     How  much  more  would  I  have  received  last 
year  for  3140  pounds  which  I  sold  this  year  at  $  .20  a  pound? 

15.  Custom  millers  take  \  of  the  quantity  of  grain  as  pay  for 
grinding  it.    How  many  bushels  must  a  man  carry  to  the  mill 
so  that  he  may  bring  back  14  bushels  of  ground  provender? 

16.  If  $45  is  f  of  my  money,  what  part  of  it  will  that  sum 
plus$4£be? 


REVIEW   EXERCISES.  127 

17.  A  farmer  had  two  fields  in  which  he  kept  his  sheep. 
In  one  there  was  -|  of  the  whole  number  of  sheep,  and  in  the 
other  there  were  148  sheep.     How  many  sheep  had  he? 

18.  A  merchant  exchanged  21  barrels  of  flour  worth  $7-f  a 
barrel,  for  24f  cords  of  wood.     What  did  the  wood  cost  him 
per  cord? 

19.  If  I  give  A  £  of  my  money,  B  £  of  it,  and  C  £  of  it, 
what  part  of  my  money  have  I  left? 

20.  Mr.  A.  owns  f  of  a  vessel  valued  at  $18326.     If  he 
sells  f  of  his  share  to  Mr.  B.,  what  part  of  the  whole  will 
he  have  left?     What  part  will  Mr.  B.  have?     What  is  the 
value  of  Mr.  A.'s  share?     Of  Mr.  B.'s  share? 

21.  After  buying  a  suit  of  clothes  for  $60  I  found  I  had  f 
of  my  money  left.     How  much  had  I  at  first? 

22.  A  man  sold  \  of  3£  cords  of  wood  for  f  of  $8|.     How 
much  did  he  receive  for  it  per  cord? 

23.  How  many  tons  of  hay  will  be  required  to  keep  7 
horses  for  6  months,  if  9  horses  eat  16^  tons  in  that  time? 

24.  If  f  of  a  farm  is  sold  for  $8516,  what  would  be  the 
worth  of  the  whole  at  the  same  rate  ? 

25.  A  gentleman  spent  \  of  his  annual  income  traveling, 
and  \  of  the  remainder  in  the  purchase  of  books.     The  rest, 
which  was   $8526,  he  expended   upon  paintings  and  other 
works  of  art.     What  was  his  annual  income? 

26.  Two  men  dug  a  ditch  for  $53;  one  man  worked  3|- 
days  and  dug  14|-  rods;  the  other  worked  as  many  days  as 
the  first  dug  rods  per  day.     How  much  did  each  receive,  if 
they  shared  in  proportion  to  the  time  they  worked? 

27.  Two  brothers  together  own  \  of  a  flouring  mill  valued 
at  $13000.     One  owns  -f  as  much  as  the  other.     What  is 
the  value  of  each  one's  share? 

28.  The  loss  caused  by  a  fire  was  $3865.     The  sum  was 
paid  by  an  insurance  company  which  insured  the  stock  for  -f 
of  its  value.     What  was  the  entire  value  of  the  stock? 


128  COMMON    FRACTIONS. 

29.  A  can  do  a  piece  of  work  in  10  days.     What  part  of  it 
can  he  do  in  1  day  ?     If  B  can  do  the  same  piece  of  work  in 
8  days,  what  part  of  it  can  he  do  in  1  day  ?     What  part  can 
both  together  do  in  1  day?     How  many  days  would  be  re- 
quired for  both  to  do  the  work? 

30.  If  A  can  do  a  piece  of  work  in  5  days  and  B  in  8  days, 
how  long  will  it  take  both  to  do  it? 

31.  If  A  and  B  can  do  a  piece  of  work  in  10  days,  and  A 
can  do  it  alone  in  15  days,  how  long  will  it  take  B  to  do  it? 

32.  A  tree  124  feet  high  was  broken  in  two  pieces  by  falling, 
f  of  the  length  of  the  shorter  piece,  equaled  y  of  the  length 
of  the  longer  piece.      What  was  the  length  of  each  piece? 

33.  A  man  who  had  spent  %  his  money  and  $-|-  more,  found 
that  he  had  $21  left.     How  much  money  had  he  at  first? 

34.  A,  B  and  C  can  do  a  piece  of  work  in  9  days.     A  can 
do  it  in  25  days,  and  B  in  30  days.     In  what  time  can  C 
do  it? 

35.  Two  ladders  will  together  just  reach  the  top  of  a  build- 
ing 75  feet  high.     If  the  shorter  ladder  is  |-  the  length  of  the 
longer  one,  what  is  the  length  of  each? 

36.  There  are  two  numbers  whose  sum  is  140,  one  of  which 
is  f  the  other.     What  are  the  numbers? 

37.  In  1875  a  merchant's  profits  were  ^  of  his  receipts;  in 
1876  they  decreased  -|,   which  diminished  his  profits  ^  of 
$2756i.     What  were  his  receipts  in  1875? 

38.  A  and  B  together  had  $5700.     |   of  A's  money  was 
equal  to  f  of  B's.     How  much  had  each? 

39.  A  man  engaged  to  work  a  year  for  $240  and  a  suit 
of  clothes.     At  the  end  of  9  months  an  equitable  settlement 
was  made  by  giving  him  $168  and  the  suit  of  clothes.    What 
was  the  value  of  the  clothes? 

40.  A  and  B  can  do  a  piece  of  work  in  12  days.     Assum- 
ing that  A  can  do  f-  as  much  as  B,  how  long  will  it  take  each 
to  do  it? 


DECIMAL  FRACTIONS 


194.  1.  If  an  apple  be  divided  into  ten  equal  parts,  what 
part  of  the  apple  will  one  of  these  parts  be?  Two  parts? 
Five  parts? 

2.  If  one-tenth  of  an  apple  be  divided  into  ten  equal  parts, 
what  part  of  the  apple  will  one  of  these  parts  be?    Two  parts? 
WhatpartoflisTVofTy    -&ofTV?    T8oOfTV 

3.  If  one-hundredth  of  a  dollar  be  divided  into  ten  equal 
parts,  what  part  of  a  dollar  will  one  of  these  parts  be  ?    Eight 
parts?     What  part  of  1  is  -^  of  y-J^? 

4.  What  part  of  one-tenth  is  one-hundredth  ?     Of  2  tenths 
is  2  hundredths?     Of  3  tenths  is  3  hundredths  ?     Of  9  tenths 
is  9  hundredths? 

5.  What  part  of  one-hundredth  is  one-thousandth  ?     Of  2 
hundredths  is  2  thousandths?     Of  8  hundredths  is  8  thou- 
sandths ? 

6.  What  are  the  divisons  of  any  thing  into  tenths,  hun- 
dredths, thousandths,  ten-thousandths,  etc.,  called? 

Atis.  Decimal  divisions. 


DEFINITIONS 

195.  A  Decimal  Fraction  is  one  or  more  of  the 
decimal  divisions  of  a  unit. 

The  word  decimal  is  derived  from  the  Latin  word  decent, 

which  signifies  ten. 

9  (129) 


130 


DECIMAL  FRACTIONS. 


Decimal  fractions,  for  the  sake  of  brevity,  are  usually  called 
decimals. 

196.  Since  tenths  are  equal  to  ten  times  as  many  hun- 
dredths,  and  hundredths  are  equal  to  ten  times  as  many 
thousandths,  thousandths  to  ten  times  as  many  ten-thou- 
sandths, etc.,  it  is  evident  that  decimals  have  the  same  law 
of  increase  and  decrease  as  integers,  and  that  the  denominator 
may  therefore  be  indicated  by  the  position  of  the  figures. 

According  to  the  decimal  system  of  notation,  figures  de- 
crease in  tenfold  ratio  in  passing  from  left  to  right;  therefore 
a  figure  at  the  right  of  units  will  express  tentlis,  at  the  right  of 
tenths,  hundredths,  at  the  right  of  hundredtlis,  tJiousandths,  etc. , 
as  is  exhibited  by  the  following  expressions : 


a   g  *j      3   o   § 

s     8     a         2     a    jg 
W    ti    P        H    B    H 

234 
147 
313 

6 

05 
005 

-234ft 

=  147y% 

164 

034 

=    164yf^- 

1  64 

03400 

—  164T-|f 

01  64 

—          _LJL 
—            inn 

From  this  mode  of  expressing  decimal  fractions  the  follow- 
ing principles  are  deduced : 

197.  PRINCIPLES. — 1.  Decimals  conform  to  the  same  prin- 
ciples of  notation  as  integers. 

2.  Each  decimal  cipher  prefixed  to  a  decimal  diminishes  its 
value  tenfold,  since  it  removes  each  figure  one  place  to  the  right. 

3.  Annexing  ciphers  to  a  decimal  does  not  alter  its  value,  since 
it  does  not  change  the  place  of  any  figure  of  the  decimal. 

4.  The  denominator  of  a  decimal,  when  expressed,  is  1  with  as 
many  ciphers  annexed  as  tliere  are  orders  in  the  decimal. 


DECIMAL  FRACTIONS.  131 

The  Decimal  Point  is  a  period  placed  before  the 
decimal.  Thus,  .6  represents  T6^;  .54  represents  -ffa* 

The  decimal  point  is  also  called  the  Separatrix,  since  it  is 
also  used  to  separate  integers  from  decimals. 

198.  A  Pure  Decimal  Number  is  one  which  con- 
sists of  decimals  only;  as  .387. 

199.  A  Mixed  Decimal  Number  is  one  which  con- 
sists of  an  integer  and  a  decimal;  as  46.3,  which  is  equal  to 
46^. 

200.  A  Complex  Decimal  is  one  which  has  a  com- 
mon fraction  at  the  right  of  the  decimal;   as  .3-f,  which  is 


. 

equal  to  3. 
10 


02. 

NUMEKATION  TABLE. 


03  CO 

^  rf  ^  H 

cj  n  Sc  93 

02  i-M  O2  •         O 


02 

O 
£ 
'd 

2 
T5 

O 

02 

T5 
C 
c5 

02 

»-j 

^ 

'•d 

.   _«• 

02 

dredths. 

isandths. 

thousand 

O 

•3 

T3 

£ 

T5 

02 

rg 
J 

millionth 

1 

13 

S 

T3 

02 

rg 
J 

IW 

rg 

1 

IB 

1 

g 

w 

fl 

O 

S 

s 

w 

1  1 

"S 

p 

M 

i—- 
O 

H 

1 

C 

W 

1 

i 

O 

S 
M 
W 

H 

i 

1 

2 

7 

5 

4 

8 

3    4    . 

6 

8 

4 

7 

3 

0 

2 

4 

2 

5 

/ 

INTEGERS.  DECIMALS. 

By  examining  this  table  it  will  be  seen  that  tenths  occupy 
the  first  decimal  place,  hundredth  the  second,  thousandths  the 
third,  ten-thousandths  the  fourth,  hundred-thousandths  the  fifth, 
millionths  the  sixth,  etc.  Hence, 

The  place  occupied  by  any  order  of  decimals  is  one  less  than 
that  occupied  by  the  corresponding  order  of  integers. 


132 


DECIMAL   FRACTIONS. 


201.  What  order  of  decimals  occupies 


1st  place? 
4th  place? 
3d  place? 
7th  place? 

5th  place? 
2d  place? 
6th  place? 
10th  place? 

9th  place? 
8th  place? 
2d  place? 
3d  place? 

What  decimal  place  is  occupied  by  hundredths?  Tenths? 
Hundred-million ths?  Thousandths?  Ten-thousandths?  Ten- 
millionths  ?  Millionths  ?  Billionths  ?  Hundred-thousandths ? 


EXAMPLES  IN  NUMERATION. 

202.    1.  Read  the  decimal  4.246. 

ANALYSIS. — The  figures  of  the  decimal  express  2  tenths,  4  hun- 
dredths and  6  thousandths,  which,  reduced  to  equivalent  fractions 
having  a  common  denominator,  become  200  thousandths,  40  thou- 
sandths and  6  thousandths,  or  246  thousandths. 

The  whole  expression  is  read  4  and  246  thousandths. 

RULE. — Read  the  decimal  as  an  integral  number  and  give  it 
the  denomination  of  the  right-hand  figure. 


Read  the  following: 

2.    .684. 

14.    .6231. 

26.    4.16. 

3.    .084. 

15.    .4896. 

27.    5.8406. 

4.    .004. 

16.    .3893. 

28.    .60000. 

5.    6.839. 

17.    18.468. 

29.    .00006. 

6.    68.39. 

18.    23.8009. 

30.    .40508. 

7.    683.9. 

19.    649.3804. 

31.    40.0004. 

8.    .00450. 

20.    .0020064. 

.     32.    4000.004. 

9.    3,02304. 

21.    .4120465. 

33.    518.6800. 

10.    .050600. 

22.    6.932474. 

34.    4000.129. 

11.    4.00008. 

23.    2.234006. 

35.    80000.86. 

12.    .000000856. 

24.    3.000600. 

36.    8000.086. 

13.    1.000003894.        25.    4.006006. 

37.    800.0086. 

NOTATION.  133 

EXAMPLES  IN  NOTATION. 

203.    1.  Express  decimally  forty-three  thousandths. 

ANALYSIS. — Since  43  thousandths  are  equal  to  4  hundreclths  and  3 
thousandths,  we  write  3  in  thousandths'  place  and  4  in  hundredths' 
place,  and  as  there  are  no  tenths,  0  in  tenths'  place.  Hence,  forty-three 
thousandths— .043. 

RULE. —  Write  the  numerator  of  the  decimal,  prefix  cipher*  if 
necessary  to  indicate  the  denominator,  and  place  the  decimal  point 
be/ore  tenths. 

Express  decimally: 

2.  Eight  tenths.     Nine  tenths.     Five  tenths. 

3.  Two  hundreclths.    Eight  hundreclths.    Six  hundredths. 

4.  Six  thousandths.     Four-hundrecl-two  thousandths. 

5.  Nine  ten-thousandths.    Eight  hundred  ten-thousandths. 

6.  Seventeen  hundreclths.     Fifteen  hundred-thousandths. 

7.  Forty-eight  thousandths.     Five  hundred  ten-millionths. 

8.  Ninety-three  ten-thousandths.     Ninety-three  billionths. 

9.  Fifty-one  hundred-thousandths.    Fifty-one  thousandths. 

10.  107  millionths.     306  ten-millionths. 

11.  3259  hundred-thousandths.     4268  hunclrecl-millionths. 

12.  429  ten-millionths.     3842  hundred-thousandths. 

13.  4300  billionths.     38496  hundrecl-billionths. 

14.  85  hundred-millionths.     85  hundrecl-billionths. 

15.  Six  thousand  ten-thousandths.     Five  ten-billion ths. 

16.  Five  and  six-tenths.     Eight  and  nine  ten-thousandths. 

17.  Eighty  ten-thousandths.     Forty  hundred-thousandths. 


Express  decimally : 
18. 


20- 


21.  4ft. 


22. 


•    i  o  o  o  • 


25. 


26. 

27. 


.     10  0* 


0  60- 


29.  T 


134  DECIMAL    FRACTIONS. 

In  reading  expressions  of  United  States  currency,  the  cents, 
mills,  etc.,  may  be  read  as  decimals  of  a  dollar.  Thus, 
$4.7235  may  be  read  4  dollars  72T3o5Q-  cents,  or 


REDUCTION. 

CASE  I. 

204.  To  reduce  dissimilar  decimals  to  similar  deci- 
mals. 

1.  How  many  tenths  of  an  apple  are  there  in  1  apple  ?    How 
many  hundredths  in  10  apples?     How  many  thousandths? 

2.  How  many  hundredths  are  there  in  6  tenths  ?     How 
many  thousandths?     How  many  ten-thousandths? 

3.  Express   6  hundredths  as  thousandths.     As  ten-thou- 
sandths.    As  hundred-thousandths.     As  millionths. 

4.  Express  8  thousandths  as  ten-thousandths.     As  hun- 
dred-thousandths.    As  millionths. 

205.  PRINCIPLE. — Annexing  ciphers  to  a  decimal  does  not 
alter  its  value. 

WRITTEN    EXERCISES. 

1.  Eeduce  .5,  .36,  .406  and  3.3109  to  similar  fractions. 

PROCESS.  ANALYSIS. — The  lowest  order  of  deci- 

5  __.       5000         nials  in  the  given  numbers  is  ten-thou- 

sandths,  and  to  reduce  the  decimals  to 

similar  decimals,  we  must  change  them 

.406     =     .4060         all  to  ten-thousandths,  or  to  other  deci- 
3   3109  =  3   3109         nials  having  an  equal  number  of  places. 

Since  annexing  ciphers  to  a  decimal 

does  not  alter  its  value,  we  give  to  each  number  four  decimal  places 
by  annexing  ciphers,  and  this  renders  them  similar. 

RULE. — Give  to  all  the  given  decimals  the  same  number  of 
decimal  places  by  annexing  ciphers. 


DEDUCTION.  135 

2.  Reduce  .6,  .75,  .089,  to  similar  decimals. 

3.  Keduce  .15,  .0406,  .0035,  .051,  to  similar  decimals. 

4.  Keduce  .0045,  .3846,  .51,  .51040,  to  similar  decimals. 

5.  Reduce  3.35,  .345,  to  similar  decimals. 

Reduce  the  following  dissimilar  decimals  to  similar  decimals  : 


6.  .0436,  .04506,  .82. 

7.  .05,  4.825,  3.6046. 

8.  .3854,  .729,  8.053. 

9.  .8104,  .0008,  8000.4. 


10.  *5,  .5,  .005,  50. 

11.  3.5,  .416,  .34,  14. 

12.  .214,  8.3,  .8,  4.6. 

13.  8.1,  43,  .68,  3.90. 


CASE  II. 
206.  To  reduce  a  decimal  to  a  common  fraction* 

1.  If  5  tenths  be  written  as  a  common  fraction  what  will 
be  the  numerator?     What  will  be  the  denominator? 

2.  AVhat  is  the  numerator  and  what  the  denominator  of 
the  decimal  18  hundredths,  when  expressed  as  a  common 
fraction  ? 

3.  Express  the  value  of  the  decimal  50  hundredths,  by  a 
common  fraction  in  its  smallest  terms. 

4.  Express  by  a  common  fraction  in  its  smallest  terms,  the 
following  decimals :  20  hundredths.    30  hundredths.    50  hun- 
dredths.    250  thousandths.     375  thousandths. 

WRITTEN    EXERCISES. 

1.  Reduce  .75  to  its  equivalent  common  fraction. 

PROCESS.  ANALYSIS. —  .75  expressed    as    a   common 

7  5  _  _7_5   __  3          fraction  is  T^\,  which,  being  reduced  to  its 
smallest  terms,  equals  f. 

RULE. — Omit  the  decimal  point,  supply  the  denominator,  and 
reduce  the  fraction  to  its  lowest  terms. 


136 


DECIMAL   FRACTIONS. 


Reduce  the  following  decimals  to  equivalent  common  frac- 
tions in  their  smallest  terms: 


2.  .054. 

6.  4.0125. 

10.  .354. 

14.  .5675. 

3.  .03875. 

7.  .4355. 

11.  .00625. 

15.  3.216. 

4.  .05625. 

8.  .0005. 

12.  .05375. 

16.  .4824. 

5.  .4375. 

9.  .5000. 

13.  .06506. 

17.  .005396. 

18.  Reduce  .15^  to  an  equivalent  common  fraction. 

PROCESS.  ANALYSIS. — The  expres- 

1  C  I 


Too 


i£.    Keducingthedenom- 


inator  also  to  sevenths,  the  expression  becomes  J$§,  or  /53Tr. 

Change  the  following  to  equivalent  common  fractions,  or  to 
mixed  numbers  : 


19.  .1%   - 
20.  .33$. 
21.  .16|. 

22.  .87$. 
23.  .04f. 
24.  .037$. 

25.  .562$. 
26.  .003|. 
27.  .078^. 

28.  .0003$. 
29.  2.756$. 
30.  13.8i|. 

f  ? 


f  ? 


CASE  III. 
207.  To  reduce  a  common  fraction  to  a  decimal. 

1.  One  half  of  an  apple  is  equal  to  how  many  tenths  of  an 
apple  ? 

2.  How  many  tenths  are  there  in    -J-?    -f  ?    f  ? 

3.  How  many  hundredths  are  there  in 

4.  How  many  hundredths  are  there  in 

5.  How  many  hundredths  are  there  in 
divided  by  2?     How  many  in  J? 

6.  How  many  hundredths  are  there  in  f-,  or  400  hundredths 
divided  by  5?     How  many  in  f  ? 

7.  How  many  thousandths  are  there  in  -f  ,  or  5000  thou- 
sandths divided  by  8?     How  many  in  f  ?     How  many  in  £? 


f  ? 


or  100  hundredths 


DEDUCTION. 


137 


.625     Or, 


WRITTEN    EXERCISES. 

1.  Reduce  -f  to  an  equivalent  decimal. 

PROCESS.  ANALYSIS. —  f  is  \  of  5,  or  50 

8)5    000  tenths ;   and  J  of  50  tenths  is  6 

tenths  anc^  2  tenths  remaining. 
2  tenths  are  equal  to  20  hun- 
dredths, and  |  of  20  hundredths 
is  2  hundredths  and  4  hundredths 
remaining.  4  hundredths  are  equal  to  40  thousandths,  and  J  of  40 
thousandths  is  5  thousandths.  Hence  f  is  equal  to  6  tenths  +  2  hun- 
dredths +  5  thousandths,  or  .625. 

Or  we  may  multiply  both  terms  of  the  fraction  by  1000  and  divide 
the  resulting  terms  by  8,  and  obtain  the  decimal  '-fffat  or  -625. 

RULE. — Annex  ciphers  to  the  numerator  and  divide  by  the  de- 
nominator. Point  off  as  many  decimal  places  in  the  quotient  as 
there  are  ciphers  annexed. 

In  many  cases  the  division  is  not  exact.  In  such  instances  the  re- 
mainder may  be  expressed  as  a  common  fraction,  or  the  sign  -f-  may  be 
employed  after  the  decimal  to  show  that  the  result  is  not  complete; 
thus:  $  =  ,166f,  or  .166  +. 

Reduce  the  following  to  equivalent  decimals : 


2.    i. 

8-  'A- 

14.    f. 

20.    A. 

3.   f. 

9.    A- 

15.    A- 

21-  if*- 

4.    f. 

10.    tf. 

16.    &. 

22.     f. 

5.    f. 

11.    «. 

17.    &. 

23.    ^. 

6.    f. 

12.    «. 

18.    ff. 

24.    TV 

7.    f. 

13.    ff 

19-  if*- 

25.    A- 

Change  the  following  to  the  decimal  form: 


26.  15f. 
27.  24|. 
28.  .821 

29.  3.4£. 
30.  .23f. 
31.  .62^. 

32.  .871 
33.  .43f. 
34.  4.21|. 

35.  37.5TV 
36.  20.  Of. 
37.  .OOQtf. 

138  DECIMAL    FKACTIONS. 


ADDITION. 

208.  1.  What  is  the  sum  of  •&  and  •&?     •&  and  •&?     .3 
and  .7? 

2.  What  is  the  sum  of  T\OQ-  and  yfo?     ^  and  -$y     .12 
and  .20? 

3.  What  is  the  sum  of  T^  and  T1IVo  ?     TWIT  and  ^jfo? 
.005  and  .043? 

4.  Find  the  sum  of  T2^  and  T|^.     Of  .5  and  .06.     .7  and 
.19. 

5.  Find  the  sum  of  .6,  .31,  .004.     Of  .5,  .08  and  .006. 

209.  PRINCIPLES. — The  principles  are  the  same  as  for  ad- 
dition of  integers. 

WRITTEN    EXERCISES. 

1.  What  is  the  sum  of  .36,  2.136  and  4.5004? 

PROCESS.  ANALYSIS. — We  write  the  numbers 

g  g        ___       3600        so  that  units  of  the  same  order  shall 

o    1  Q  £      _o    1  Q  £  ft         stand  in  the  same  column,  and  add  as 
L  .  1  o  0      —  Z.loOU          •'    •   .  A.       ,11-       i 

A    K(\(\  A        A    ^f\(\  A         in  mtegers>  separating  the  decimal  part 
4'5UU4  ==  4'5004        of  the  sum  from  the  integral  part  by 
6    9964       6    9964        ^e  Decimal  point.     The  decimals  are 
made  similar  by  annexing  ciphers  until 

all  the  decimals  have  the  same  number  of  places. 

It  is  not  usual  to  make  the  decimals  similar,  for  if  they  are  written 

so  that  decimals  of  the  same  order  stand  in  the  same  column  it  is 

unnecessary  to  supply  the  ciphers. 

RULE. — The  rule  is  the  same  as  for  addition  of  integers. 
What  is  the  sum  of 


2.  4.15,  3.86  and  .487? 

3.  3.9,  4.84  and  .0507? 

4.  .004,  5. 86  and  3. 05? 


5.  6.843,  48.25  and  17.286? 

6.  .35, .046  and  .00435? 

7.  106,  .106,  1.06  and  10. 6? 


SUBTRACTION.  139 

8.  What  is  the  sum  of  $5.18,  $3.09,  $46.  and  $54.185? 

9.  Find  the  sum  of  $18.23,  $12.08,  $31.255  and  $6.625. 

10.  Add  $34.73,  $206.357,  $1200.18,  $3816  and  $137. 

11.  Express  as  decimals  and  add  6J,  3f,  5f,  6i  and  9f. 

12.  A  laborer  earned  $7.25  in  one  week,  $7.12^  in  another, 
$9.18f  in  another,  and  $8f  in  another.     How  much  did  he 
earn  during  that  time? 

13.  What  is  the  sum  of  18  thousandths,  15  millipnths,  81 
hundredths,  146  ten-thousandths,  834  hundred- thou sandths  ? 

14.  What  is  the  sum  of  8  dollars  5  cents,  13  dollars  19 
cents,  18  dollars  3  cents  8  mills,  25  dollars  37  cents  5  mills, 
12f  dollars,  and  -^  of  a  dollar? 

15.  Mr.  A.  paid  the  following  bills  for  repairs  upon  his 
premises,  viz :  carpenter-wrork,  $381.45;  plastering,  $215.385; 
plumbing,  $323.94;  and  other  expenses,  $181.57.    How  much 
did  he  pay  for  repairs  ? 

16.  A  farmer  purchased  cloth  for  $13f-,  boots  for  $8^-, 
crockery  for  $10{^,  and  groceries  for  $15.49.     How  much 
did  he  pay  for  all  his  purchases  ? 


SUBTRACTION. 

210.  1.  From  T5o  take  TV     From  .9  take  .5. 

2.  Find  the  difference  between  T-|-Q  and  yf^;  -£fo  and 
.19  and  .08. 

3.  Find  the  difference  between  y^Vfr  an(^  ToVo>'  TTOTF 
Y^-;  .007  and  .005. 

4.  What  is  the  difference  between  -£$  and  y|~o?     .5  and 
.06?     .7  and  .09? 

5.  What  is  the  difference  between  .16  and  .03?     .15  and 
.08?     .45  and  .3? 

211.  PRINCIPLES. — The  principles  are  the  same  as  for  the 
subtraction  of  integers. 


140  DECIMAL    FRACTIONS. 


WRITTEN    EXERCISES. 

I.  From  34.634  take  5.6857. 

PROCESS.  ANALYSIS. — We  write  the  numbers  so  that  units 

34.6340         °f  tne  same  order  shall  stand  in  the  same  column, 

5.6857         and  subtract  as  in  integers,  separating  the  decimal 

28    9  4  8  3  Part  °^  ^ie  rema^n(^er  fr°m  tne  integral  part  by  a 

decimal  point. 

Or,  .  In  the  first  process  the  decimals  are  made  simi- 

~       P  o  <  lar  ^7  annexing  a  cipher  to  the  minuend. 
_  *     ~  r  „  In  the  second  process,  which   is   the  one  com- 

- —  monly  employed,  the  cipher  is  not  written,  but  we 

28.9483  suppose  it  to  be  annexed. 

RULE. — The  rule  is  the  same  as  for  subtraction  of  integers. 

(2.)  (3.)  (4.)  (5.) 

From        48.356        39.82          $43.25          $118.375 
Subtract  23.453         13.856        $18.375        $  43.50 

6.  What  is  the  difference  between  .7134  and  .50645? 

7.  What  is  the  difference  between    8.34  and  6.3168? 

8.  What  is  the  difference  between     100  and  .03846? 

9.  From  84  millionths  take  84  ten-millionths. 
10.  From  80  thousand  take  80  thousandths. 

II.  From  29  dollars  3  cents  take  17  dollars  9  cents. 

12.  From  27  dollars  8  cents  take  9  dollars  37  cents  5  mills. 

13.  If  I  spend  $45. 89^-  for  merchandise,  how  much  will  I 
have  left  after  paying  for  it  with  a  fifty-dollar  bill  ? 

14.  A  gentleman's  income  wras  $12384.16,  and  his  expenses 
the  same  year  were  $9864.18.     How  much  of  his  income  was 
left? 

15.  The  receipts  of  a  reaper  manufactory  for  the  year  1876 
were  $1374837.64,  and  the  expenditures  $1298369.58.    What 
was  the  surplus  ? 


MULTIPLICATION.  141 


MULTIPLICATION. 

212.  1.  What  is  the  product  of  •&  X  2  ?    -&  X  3  ?    .4X2? 

2.  How  many  decimal  figures  are  there  in  the  product  of 
tenths  by  units? 

3.  What  is  the  product  of  -fa  X  4  ?    yf ^  X  4  ?    .04  X  2  ? 

4.  How  many  decimal  figures  are  there  in  the  product  of 
hundredths  by  units? 

5.  What  is  the  product  of  T2¥XT3o  ?    fVXTV     .4  X  .2? 

6.  How  many  decimal  places  are  required  to  express  the 
product  of  tenths  multiplied  by  tenths  ? 

7.  What  is  the  product  of  ^  X  Tfo ?    TBO  X  Tth) ?    -4  X  .02? 

8.  How  many  decimal  places  are  required  to  express  the 
product  of  tenths  multiplied  by  hundredths  ?    Tenths  by  thou- 
sandths ?     Hundredths  by  thousandths  ? 

9.  If  the  multiplicand  has  two  decimal  places,  and  the  mul- 
tiplier three,  how  many  will  there  be  in  the  product  ? 

10.  How  does  the  number  of  places  required  to  express  the 
product  of  two  decimals  compare  with  the  number  of  decimal 
places  in  the  factors? 

213.  PRINCIPLE. — The  product  of  two  decimals  contains  as 
many  decimal  places  as  there  are  decimal  places  in  both  factors. 

WRITTEN    EXERCISES. 

1.  Multiply  .312  by  .24. 

PROCESS.  ANALYSIS.—  .31 2  X  -24  =  J$fo  X  T2o4<3  =  iU §§o  = 

.312        -07488.    Hence  .31 2  X  .24  =  .07488.    Or, 

9  4  We  may  multiply  as  if  the  numbers  were  integers; 

and  since  the  multiplicand  has  three  decimal  places, 

1248         an(]  the  multiplier  two  places,  the  product  must  have 

624  five  places.     (Prin.)     Or,  thousandths  multiplied  by 


07488         hundredths  give  hundred-thousandths,  the  denomi- 
nation of  the  product. 


142 


DECIMAL    FRACTIONS. 


RULE. — Multiply  as  if  the  numbers  were  integers,  and  from  the 
right  of  the  product  point  off  as  many  figures  for  decimals  as  there 
are  decimal  places  in  both  factors. 

If  the  product  does  not  contain  as  many  figures  as  there  are  decimals 
in  both  factors  the  deficiency  must  be  supplied  by  prefixing  ciphers. 


Multiply: 

Multiply  : 

2.     .65  by  .34. 

14.       $2.75  by  8|. 

3.     .45  by  4.5. 

15.     $31.16  by  5|. 

4.    .436  by  .46. 

16.     34.165  by  3f. 

5.     348  by  .44. 

17.       3.845  by  7.3. 

6.   3.48  by  .64. 

18.     $15.18  by  .666f. 

7.    34.8  by  .74. 

19.     500.15  by  5.36. 

8.   8.75  by  8.5. 

20.     37.856  by  30.04. 

9.    .579  by  .035. 

21.       70.05  by  .0405.     . 

10.     486  by  3.75. 

22.       63.18  by  2.308. 

11.   2.48  by  2.37. 

23.    -64.032  by  .0634. 

12.   3.94  by  3.84. 

24.      51.27  by  5.321. 

13.  5384  by  .0064. 

25.  |  of  .55  by  £  of  6.5. 

26.  Multiply  4.639  by  100. 


PROCESS. 

4.639 

100 
463.900 


ANALYSIS. — Since  each  removal  of  a  figure  one 
place  to  the  left  increases  its  value  tenfold,  the 
removal  of  the  decimal  point  one  place  to  the  right 
multiplies  by  10,  and  the  removal  of  the  point  two 
places  to  the  right  multiplies  by  100.  Hence, 


RULE. — To  multiply  by  1  with  any  number  of  ciphers  annexed, 
remove  the  decimal  point  as  many  places  to  the  right  as  there  are 
ciphers  annexed. 


27.  Multiply  384.64  by  100.     By  10.     By  1000. 

28.  Multiply  1.8465  by  100.     By  1000.     By  10000. 

29.  What  will  be  the  cost  of  34.5  yards  of  cloth  at 
per  yard? 


3.15 


DIVISION.  143 

30.  When  land  is  worth  $137.18  per  acre,  how  much  must 
be  paid  for  a  farm  of  38  acres? 

31.  Since  16.5  feet  make  a  rod,  how  many  feet  are  there 
in  23.7  rods? 

32.  What  will  be  the  cost  of  9  houses  at  $3847.93  each? 

33.  When  wheat  is  worth  $1.62^  per  bushel,  what  will  37.3 
bushels  cost? 

34.  What  is  the  value  of  57  barrels  of  flour  at  $8.37£  a 
barrel ? 

35.  Mr.  Orr  sold  8  horses  at  an  average  price  of  $213.27 
each.     How  much  did  he  receive  for  them? 

36.  A  lady  made  the  following  purchases,  viz:  37  yards  of 
bleached  sheeting  at  $  .13^  per  yard,  8  yards  of  velvet  ribbon 
at  $  .37|  per  yard,  27  yards  of  silk  at  $2.35  per  yard.    What 
was  the  entire  cost  of  her  purchases  ? 


DIVISION. 

214.    1.  What  is  the  product  of  .6  by  8? 

2.  4.8  is  the  product  of  two  factors,  one  of  which  is  8: 
what  is  the  other  factor? 

3.  What  is  the  product  of  .6  by  .8? 

4.  .48  is  the  product  of  two  factors,  one  of  which  is  .6: 
what  is  the  other  factor? 

5.  What  is  the  product  of  .06  by  .8? 

6.  .048  is  the  product  of  two  factors,  one  of  which  is  .06* 
what  is  the  other  factor? 

7.  What  is  the  product  of  .06  by  .08? 

8.  .0048  is  the  product  of  two  factors,  one  of  which  is  .06: 
what  is  the  other  factor? 

9.  How  many  decimal  places  are  there  in  the  quotient  when 
tenths  are  divided  by  units  ?    Hundredths  by  tenths  ?    Thou- 
sandths by  hundredths  ?     Ten-thousandths  by  hundredths  ? 


144  DECIMAL   FRACTIONS. 

10.  How  many  decimal  places  are  there  in  the  product  of 
any  two  factors? 

11.  If  the  product  and  one  of  two  factors  are  given,  how 
may  the  number  of  decimals  in  the  other  factor  be  found  ? 

12.  How  may  the  factor  be  found? 

13.  Since  the  factor  sought  will  be  the  quotient,  how  many 
decimal  places  will  there  be  in  the  quotient? 

215.  PRINCIPLE. — The  quotient  will  contain  as  many  deci- 
mal places  as  the  number  of  decimal  places  in  the  dividend  ex- 
ceeds those  in  the  divisor. 


WRITTEN    EXERCISES. 

1.  Divide  8.88  by  2.4. 

PROCESS.  ANALYSIS.  —8.88  -~  2.4  —  f  J  J  -*-}$  =  f  ft 

2.4)8.88(3.7         XJJ  =  tt  =  3.7^  Or, 

7  2  We  divide  as  if  the  numbers  were  inte- 

gers ;  and  since  the  dividend  has  two  deci- 
mal places,  and  the  divisor  one,  the  quotient 
will  have  one.  (Prin.) 

RULE. — Divide  as  if  the  numbers  were  integers,  and  from  the 
right  of  the  quotient  point  off  as  many  figures  for  decimals  as  the 
number  of  decimal  places  in  the  dividend  exceeds  the  number  of 
those  in  the  divisor. 

1.  If  the  quotient  does  not  contain  a  sufficient  number  of  decimal 
places  the  deficiency  must  be  supplied  by  prefixing  ciphers. 

2.  Before  commencing  the  division,  the  number  of  decimal  places 
in  the  dividend  should  be  made  at  least  equal  to  the  number  of  deci- 
mal places  in  the  divisor. 

3.  When  there  is  a  remainder  after  using  all  the  figures  of  the  divi- 
dend, annex  decimal  ciphers  and  continue  the  division. 

4.  For  the  ordinary  purposes  of  business  it  is  not  necessary  to  carry 
the  division  further  than  to  obtain  four  or  five  decimal  figures  in  the 
quotient. 


DIVISION. 


145 


Divide  : 

Divide  : 

2. 

2.450 

by 

9.8. 

9. 

.04905 

by 

.327. 

3. 

.00335 

by 

.67. 

10. 

135.05 

by 

.037. 

4. 

6.2512 

by 

3.7. 

11. 

687.50 

by 

.025. 

5. 

.05475 

by 

15. 

12. 

34.368 

by 

.013. 

6. 

18.312 

by 

.24. 

13. 

.014582 

by 

.0692. 

7. 

105.70 

by 

3.5. 

14. 

71.142 

by 

.0071. 

8. 

.11928 

by 

.056. 

15. 

.027538 

by 

.0326. 

16.  Divide  423.68  by  100. 


PKOCESS. 

100)423.68 


ANALYSIS.— Since  each  removal  of  a  figure 
one  place  to  the  right  decreases  its  value  ten- 
fold, the  removal  of  the  decimal  point  one  place 
to  the  left  divides  by  10,  and  the  removal  of 


4.2368 
the  decimal  point  two  places  to  the  left  divides  by  100.     Hence, 


RULE. — To  divide  by  1  with  any  number  of  ciphers  annexed, 
remove  the  decimal  point  as  many  places  to  the  left  as  there  are 
ciphers  annexed. 


Divide : 

17.  48.26  by  100. 

18.  382.457  by  1000. 

19.  13.8542  by  1000. 


Divide : 

20.  4.897  by  100. 

21.  .06045  by  1000. 

22.  3845.63  by  10000. 


23.  At  $8.25  per  ton,  how  much  hay  can  be  bought  for 
$29.35? 

24.  At  $.18  per  dozen,  how  many  dozen   eggs   can   be 
bought  for  $32.40? 

25.  If  I  pay  $106.40  for  35  hats,  how  much  do  they  each 
cost  me? 

26.  How  many  hogsheads  of  molasses,   at  $57.38  each, 
can  be  purchased  for  $1319.74? 

27.  How  many  stoves,  at  $21.35  each,  can  be  bought  for 

$789.95? 
10 


146  DECIMAL   FRACTIONS. 


SHORT  PROCESSES. 

216.  Many  methods  have  been  devised  for  abbreviating 
the  processes  of  computation,  among  which  the  following  are 
of  much  practical  value : 

CASE  I. 

217.  To  multiply  by  a  number  a  little  less  than  a 
unit  or  the  next  higher  order. 

1.  How  much  less  than  10  is  9?     Than  100  is  99?     Than 
1000  is  999?     Than   100  is  98?     Than  100  is  97?     Than 
100  is  96? 

2.  How  much  less  than  10  times  a  number  is  9  times  the 
number?     Than  100  times  is  99  times?     Than  1000  times  is 
999  times?     Than  100  times  is  97  times?     Than  1000  times 
is  997  times? 

WRITTEN    EXEMCISES. 

1.  Multiply  4685  by  97. 

PKOCESS.  ANALYSIS. — Ninety-seven 

468500  =  100  times  4685       times  a  n"mber  is  100  times 

14055  =       3  times  4685      the  number  minus  3  times 

the  number.     We  annex  two 

454445=      97  times  4685       ciphers  to  the  multiplicand, 

thus  multiplying  by  100,  and 

then  subtract  from  the  product  three  times  the  multiplicand,  leaving 
97  times  the  multiplicand. 


Multiply : 


2.  3856  by  99. 

3.  4832  by  998. 

4.  48567  by  999. 

5.  89736  by  98. 


Multiply : 


6.  346725  by  997. 
.  7.  486965  by  97. 

8.  843256  by  96. 

9.  586436  by  9996. 


I 


SHOUT   PROCESSES.  147 


10.  What  will  be  the  cost  of  385  bushels  of  corn  at  $  .97 
per  bushel? 

11.  How  much  must  be  paid  for  1373  pounds  of  tea  at 
$.96  per  pound? 

12.  At  $97  per  acre,  what  will  a  farm  of  139  acres  cost? 

CASE  II. 

218.  To  multiply  when  one  part  of  the  multiplier  is 
a  factor  of  another  part. 

1.  Multiply  4  by  8;  4  by  8  tens;  4  by  16  tens. 

2.  Multiply  7  by  6;  7  by  6  tens;  7  by  18  tens. 

3.  When  7  times  a  number  is  known,  how  may  21  times 
a  number  be  found?     21  teris  times?     21  hundreds  times? 


WRITTEN    EXERCISES. 

1.  Multiply  3684  by  124. 

PROCESS.  ANALYSIS. — The  multiplicand  may  be  regarded 

3684         as  comPosed  of  12  tens  and  4  units,  or  3  times  as 

-^24        many  tens  as  units.     We  therefore  first  multiply  by 

— : 4  units;  and  since  there  are  3  times  as  many  tens 

1  A  7  Q  A 

as  units,  we  multiply  this  product  by  3,  and  write 

the  result  as  tens  by  placing  it  one  place  to  the  right 
456816         of  units- 


Multiply : 


2.  3825  by  63. 

3..  5973  by  93. 

4.  8126  by  123. 

5.  6924  by  213. 

6.  14273  by  246. 

7.  28653  by  328. 

8.  68435  by  217. 


Multiply : 

9.  3692  by  357. 

10.  6384  by  248. 

11.  4239  by  369. 

12.  12783  by  189. 

13.  36412  by  279. 

14.  36485  by  2408. 

15.  29753  by  3609! 


148 


DECIMAL,   FKACTIONS. 


an  aliquot 


Is 


121? 


CASE  III. 

219.  To  multiply  by  a  number  which 
part  of  some  higher  unit. 

1.  What  part  of    10  is    2?    Is    5?    Is  2J?     Is  3£? 

2.  What  part  of  100  is  10?     Is  20?    Is  25?    Is  50? 

Is  331? 

3.  What  part  of  a  dollar  are  50  cents?     25  cents?     20 
cents?     10  cents?     12|  cents? 

4.  What  part  of  a  dollar  are  33£  cents?     16|  cents? 

220.  An  Aliquot  Part  of  a  number  is  such  a  part  as 
will  exactly  divide  the  number. 

Thus,  5,  10,  12J,  etc.,  are  aliquot  parts  of  100. 

The  aliquot  parts  of  10,  commonly  used,  are : 
5  =  1  of  10.        |      3^1  of  10.      |     21=  \  of  10. 

The  aliquot  parts  of  100,  commonly  used,  are: 


50  =  |  of  100. 
25  =  \  of  100. 
20  =  |  of  100. 

33£  =  £  of  100. 
16f  =  |  of  100. 
12J  =  i  of  100. 

10  =  i^  of  100. 
8^  =  tL0f  100. 
6^  =  ^  of  100. 

Other  parts  of  100  are : 

40  =  |  of  100.  37|  =  |  of  100. 
60  —  f  of  100.  621  =  |  of  100. 
80  =  |  of  100.  87f  =  |  of  100. 


66|  =  |  of  100. 

75  =  f  of  100. 

431  =  ^  of  100. 


WRITTEN     EXERCISES. 

1.  Multiply  434  by  25. 


PROCESS. 


ANALYSIS. — Since  25  is  \  of  100,  we  may  mul- 
4)43400        tiply  by  25  by  first  multiplying  by  100,  and  then 
10850        taking  \  of  the  product. 


SHORT    PROCESSES.  149 

2.  What  will  85  yards  of  cloth  cost  at  $  .33|  per  yard? 

ANALYSIS.— At  $1  per  yard,  the  cloth  would 
3)85  cost  $85;  and  at  $.33J,  or  |  of  a  dollar,  it  will 

2  8    3  3  l         cost  i  of  $85>  or 


Multiply : 

3.  688  by  12|. 

4.  402  by  16|. 

5.  5056  by  25. 

6.  75630  by  33|. 


Multiply : 

7.  8404  by  50. 

8.  2160  by  37|. 

9.  4236  by  66f . 

10.    7288  by  75. 


11.  What  will  be  the  cost  of  27  yards  of  cloth  at  $  .25  per 
yard? 

12.  When  butter  is  worth  33^  cents  a  pound,  what  will 
824  pounds  be  worth? 

13.  What  will  be  the  cost  of  216  pounds  of  tea  at  $  .75  per 
pound  ? 

14.  What  will  287  bushels  of  oats  cost  at  37^-  cents  per 
bushel? 

15.  What  will  394  bushels  of  potatoes  cost  at  62|-  cents 
per  bushel? 

16.  What  is  the  value  of  319  bushels  of  wheat  at  $1.37| 
per  bushel? 

CASE  IV. 

221.  To  find   the   cost  when   the   quantity  and   the 
price  of  1OO  or  1OOO  are  given. 

1.  When  the  cost  of  100  articles  is  known,  how  can  the 
cost  of  500  be  found?     600?     800?     900? 

2.  When  the  cost  of  100  articles  is  given,  how  can  the 
cost  of  250  be  found?     350?     750?     850?     950? 

3.  How  many  times    100  is    250?       275?       280?       285? 

4.  How  many  times  1000  is  3000?     3500?     3750?     4585? 


150  DECIMAL   FRACTIONS. 


WRITTEN    EXERCISES. 

I.  What  will  be  the  cost  of  375  pounds  of  fish  at  $6.75 
per  100  pounds? 

PROCESS.  ANALYSIS. — Since  100  pounds  cost  $6.75,  375 

$  6    7  5         pounds,  or  3.75  times  100  pounds,  will  cost  3.75 

g    75         times  $6.75,  or  $25.31  {.    Or,  the  price  may  be 

multiplied  by  the  quantity,  and  the  decimal  point 

$25.3125         removed  two  places  to  the  left  in  the  product. 

The  letters  C  and  M  are  used  instead  of  the  words  hundred  and 
thousand,  respectively. 

2.  How  much  will  6075  pounds  of  coal  cost  at  $  .35  per 
hundred-weight  ? 

3.  When  shingles  cost  $4.75  per  M,  how  much  will  8609 
shingles  cost? 

4.  What  is  the  price  of  a  load  of  hay  weighing  1925 
pounds,  at  $9.50  per  ton  (2000  pounds)? 

5.  What  is  the  cost  of  16795  pounds  of  plaster  at  $4.50 
per  100  pounds? 

6.  How  much  will  129765  laths  cost  at  $2.75  per  M? 

7.  What  is  the  cost  of  6975  bricks  @  $3.25  per  M? 

8.  What  is  the  cost  of  1825  pounds  of  iron  @  $45  per  ton? 

9.  How  much  must  be  paid  for  6780  envelopes  @  $2.75 
per  M? 

10.  What  would  be  the  cost  of  550  pine-apples  at  $13.25 
per  C? 

II.  What  will  be  the  cost  of  1592  pounds  of  beef  at  $4.50 
per  hundred  pounds? 

12.  What  will  15000  pounds  of  coal  cost  at  $7.50  a  ton? 

13.  What  will  be  the  cost  of  2294  pounds  of  broom-corn 
at  $55  per  ton? 

14.  What  will  be  the  cost  of  1964  pounds  of  maple-sugar 
at  $13.45  per  hundred- weight? 


ACCOUNTS   AND   BILLS. 


151 


ACCOUNTS  AND  BILLS. 

222.  A  Debt  is  an  amount  which  one  person  owes  to 
another. 

223.  A  Credit  is  an  amount  which  is  due  to  a  person, 
or  a  sum  paid  towards  discharging  a  debt. 

224.  A  Debtor  is  a  party  owing  a  debt. 

225.  A  Creditor  is  a  party  to  whom  a  debt  is  due. 

226.  An  Account  is  a  record  of  debts  and  credits  be- 
tween parties  doing  business  with  each  other. 

227.  The  Balance  of  an  Account  is  the  difference 
between  the  amount  of  the  debts  and  credits. 

228.  A  Bill  is  a  written  statement  given  by  the  seller  to 
the  buyer,  of  the  quantity  and  price  of  each  article  sold,  and 
the  amount  of  the  whole. 

229.  The  Footing  of  a  Bill  is  the  total  cost  of  all 
the  articles. 

230.  A  bill  is  Receipted  when  the  words  Received  Pay- 
ment are  written  at  the  bottom,  and  the  creditor's  name  is 
signed  either  by  himself  or  some  authorized  person. 

231.  The  following  abbreviations  are  in  common  use: 


@, 

At. 

Do, 

The  same. 

Mdse., 

Merchandise. 

%> 

Account. 

Doz, 

Dozen. 

No, 

Number. 

Acc't, 

Account. 

Dr, 

Debtor. 

Pay't, 

Payment. 

Bal, 

Balance. 

Fr't, 

Freight, 

Pd, 

Paid. 

Bbl., 

Barrel. 

Hhd, 

Hogshead. 

Per, 

By. 

Bo't, 

Bought. 

Inst, 

This  month. 

Rec'd, 

Received. 

Co, 

Company. 

Int., 

Interest. 

Yd, 

Yard. 

Or, 

Creditor. 

Lb, 

Pound. 

Yr, 

Year. 

152 


^5) 


DECIMAL,   FRACTIONS. 


v 

N     ! 


1     \  X 

^  ^'  ^ 

^  X  Xk 

•  ^  x     \ 


^ 


y 


V- 

1 

k 


« 

SfeSi 


H 


ACCOUNTS   AND    BILLS.  153 

Copy,  fill  out  and  find  the  footings  of  each  of  the  following : 

(2.) 

KOCHESTER,  March  1,  1877. 
MR.  J.  B.  ADAMS, 

Bought  of  HOWE  &  ROGERS  : 


75  J  yards  of  Carpeting    . 
37    yards  of  Drugget 

.     .     (fy    $2.12J 
"        1  20 

$ 

8    Edge   

5    Mats                           .     . 

.     .      "       4.16 

"       2  37  ^ 

18    yards  Oil-cloth.     .     .     . 

.     .      "       1.08 

9    yards  Carpet  Lining 
3    Carpet-sweepers    .     .     . 
2    doz.  Stair-rods  .... 

.     .      «         .12J 
.     .      "        2.00 

.     .      "       8.25 

* 

Received  Payment, 

HOWE  &  ROGERS. 


(3.) 


MEMPHIS,  May  20,  1877. 


MR.  GEORGE  B.  SHERMAN, 

To  SAMUEL  B.  SMALLWOOD,  Dr. 


To  37  bbl.  Pork  ....... 

©  $24.35 

s 

"  127  bbl.  Flour.     ...... 

"       8.15 

"      3  hhd.  Molasses—  169  gal.  .     . 
"    29  firkins  Butter—  2120  Ib.  .     . 
"      3  boxes  Raisins    

.43 
.31 
"       4.65 

"      5  bbl.  Kerosene—  207  gal.  .     . 
"     25  doz.  cans  Fruit  
"      3  packages  Tobacco—  318  Ib.  . 
"     13  doz.  Spices    

.18} 

2.40 
.45 
"       1.10 

$ 

Received  Payment  by  note  at  60  days, 

SAM'L  B.  SMALLWOOD. 


154 


DECIMAL   FKACTIONS. 

(4.) 


MR.  ERASTUS  P.  GATES, 


NEW  YORK,  April  1,  1877. 
To  STUKDEVANT  &  Co.,  Dr. 


1877. 

Jan. 

9 

To  3  Gold  Watches—  $124.50,  $61.24,  $57.18 

$ 

u 

13 

"    437  pwt.  Gold  Chains     .     . 

@    $1.15 

Feb. 

3 

"      35  sets  Plated  Tea-service. 

"     43.10 

" 

15 

«          17    a            a                 a 

"     51. 

Mar. 

8 

11        5  Silver  Pie-knives     .     . 

"      12. 

u 

12 

"      12  Plated  Ice-pitchers  .     . 

"      12.50 

Or.  

$ 

1877. 

Jan. 

24 

By  Cash      

$21 

20 

Feb. 

10 

«    Draft     

327 

50 

18 

"    Mdse.  returned  

78 

67 

$ 

How  much  is  still  due  Sturdevant  &  Co.? 

Make  out  in  proper  form  and  receipt  the  following: 

5.  Mrs.  M.  T.  Dana  bought  of  G.  C.  Smith  &  Co.,  25  yd. 
of  calico  @  10  cents,  37  yd.  of  sheeting  @  18^  cents,  2  pairs 
of  gloves  @  $1.50,  1  sun-umbrella  @  $6.75,  5  yd.  of  Ham- 
burg edging  @  25  cents,  7  pairs  of  hose  @  $  .85. 

6.  Mr.  C.  C.  Lovell  bought  of  R.  P.  Lawton  7568  feet  of 
hemlock  @  $12.75  per  M,  8539  feet  of  pine  flooring  @  $23.50 
per  M,  5608  feet  of  clear  pine  @  $45  per  M,  3815  feet  of  oak 
joists  @  $32  per  M,  7346  feet  of  ash  flooring  @  $34  per  M. 

7.  Mr.  George  M.  Line  bought  of  Steele  &  Avery  15  reams 
of  commercial  note  paper  @  $1.25,  7500  envelopes  @  $3.65 
per  M,  18  gross  steel  pens  @  $  .75  per  gross,  24  Ridpath's 
Histories  @  $1.25,  9  Webster's  Dictionaries  @  $10.25. 


DECIMAL   FRACTIONS.  155 


REVIEW   EXERCISES. 

1.  A  farmer  sold  his  butter  at  34  cents  a  pound,  and 
received  for  it  $123.59.     How  many  pounds  did  he  sell? 

2.  A  gallon  of  distilled  water  weighs  8.339  pounds.    How 
much  will  15^  gallons  weigh? 

3.  A  square  rod  contains  272^  square  feet.     How  many 
square  feet  are  there  in  7|-  square  rods? 

4.  The  best  anthracite  coal  is  said  to  weigh  55.32  pounds 
per  cubic  foot.     How  many  cubic  feet  will  weigh  a  ton  of 
2000  pounds? 

5.  The  number  of  cubic  inches  in  a  bushel  is  2150.42. 
How  many  cubic  inches  are  there  in  1000  bushels? 

6.  What  is  the  quotient  when  3  is  divided  by  3  thou- 
sandths? 

7.  What  is  the  quotient  when  300  is  divided  by  3000 
hundred-millionths  ? 

8.  A  lumber  merchant  had  2182565  ft.  of  lumber.     After 
selling  .20,  or  20  per  cent.,  of  it,  he  lost  15  per  cent,  of  the 
remainder  by  fire.     How  many  feet  of  lumber  were  burned? 

9.  What  will  385  pounds  of  flour  cost  at  $4.25  per  hun- 
dred-weight? 

10.  At  $  .111  per  pound,  how  many  pounds  of  sugar  can 
be  bought  for  131.25? 

11.  Bought  26  yards  of  broadcloth  at  $4.37|  per  yard, 
and  paid  for  it  in  pork  at  $7.25  per  hundred-weight.     How 
much  pork  will  it  take  to  pay  for  the  cloth? 

12.  If  15  tons  of  hay  cost  $125.25,  what  will  35  tons  cost? 

13.  If  Ridpath's  histories  retail  at  $1.25  each,  what  will 
be  received  for  350  sold  at  that  rate? 

14.  When  pork  is  selling  at  $6.25  per  hundred-weight, 
how  much  can  be  bought  for  $325? 

15.  When  8000  is  divided  by  .004,  what  is  the  quotient? 


156  DECIMAL    FRACTIONS. 

16.  When  .0008  is  divided  by  40000,  what  is  the  quotient? 

17.  How  many  days  must  a  laborer  work  at  $1.37-^  per 
day,  to  pay  for  8  cords  of  wood  at  $4.43f  per  cord? 

18.  A  lady  bought  the  following  articles:   27  yards  of  silk 
at  $2.75  per  yard,  11  yards  of  lace  at  $6.37|  per  yard,  9 
pairs  of  gloves  at  $2.15  per  pair,  10  pairs  of  hose  at  $1.10 
per  pair.     What  was  the  amount  of  the  purchase? 

19.  If  a  man  earns  $12^-  per  week,  and  spends  $7f  per 
week,  in  how  many  weeks  can  he  save  $500? 

20.  What  is  the  value  of  95150  bricks  at  $7.25  per  M? 

21.  What  is  the  value  of  a  farm  of  195  acres  if  91  acres 
are  worth  $6688.50,  and  the  remainder  $1.12|  per  acre  more? 

22.  A  drover  bought  375  sheep  at  $4.50  per  head.     He 
sold  200  of  them  at  a  loss  of  $  .20  per  head,  and  gained 
enough  on  the  rest  to  balance  the  loss.     What  did  he  get 
per  head  for  the  rest? 

23.  The  expenses  of  conducting  a  business  enterprise  were 
.40  of  the  entire  profits.     If  the  profits  were  .15  of  the  value 
of  the  goods  sold,  how  much  was  received  from  the  sale  of 
goods  if  the  profits  were  $9000  more  than  the  expenses? 

0.    ,-,  (|-iV)X(3  +  f) 

24.  Express  as  a  decimal  (1,  + ,  }  +  (3  _  lf  }  x  5  ' 

25.  A  speculator  bought  5000  bushels  of  corn  at  $  .65  per 
bushel.     He  sold  .25  of  it  for  $  .70  per  bushel,  and  the  re- 
mainder for  such  price  that  he  realized  a  profit  on  the  whole 
of  $447.50.     How  much  did  he  get  per  bushel  for  the  re- 
mainder? 

.  26.  The  estimated  value  of  Mr.  A.'s  farm  was  $6500.  If  he 
sold  a  portion  of  it,  at  its  estimated  value  per  acre,  for  $2275, 
what  decimal  part  of  the  farm  did  he  sell? 

27.  A,  B  and  C  divide  645^-  bushels  of  wheat  among  them- 
selves. A  takes  .37^,  B  ^-,  and  C  the  remainder.  How 
many  bushels  had  each? 


DEFINITIONS. 

232.  A  Concrete  Number  is  a  number  used  in  con- 
nection with  some  specified  thing. 

Thus,  5  books,  7  trees,  8  horses,  are  concrete  numbers. 

233.  An  Abstract  Number  is  a  number  that  is  not 
used  in  connection  with  any  specified  thing. 

Thus,  5,  7,  8,  are  abstract  numbers. 

234.  A  Denominate  Number  is  a  concrete  number 
in  which  the  unit  of  measure  is  established  by  law  or  custom. 

Thus,  5  yards,  3  feet,  7  pounds,  3  ounces,  are  denominate  numbers. 

235.  A  Simple  Denominate  Number  is  a  denom- 
inate number  composed  of  units  of  the  same  denominations. 

Thus,  5  feet,  9  pounds,  3  miles,  are  simple  denominate  numbers. 

236.  A  Compound  Denominate  Number  is  a 

denominate  number  composed  of  units  of  two  or  more  denom- 
inations which  are  related  to  each  other. 

Thus,  6  feet  and  4  inches,  8  hours  and  32  minutes,  are  compound 
denominate  numbers. 

237.  A  Standard  Unit  is  a  unit  of  measure  from 
which  the  other  units  of  the  same  kind  may  be  derived. 

Thus,  the  yard  is  the  standard  unit  from  which  all  measures  of  length 

are  formed;  the  Troy  pound  the  standard  unit  of  weight. 

(157) 


158  DENOMINATE    NUMBERS. 

238.  A  Scale  is  the  ratio  by  which  numbers  increase  or 
decrease.     Scales  are  either  uniform  or  varying. 


MEASURES  OF  VALUE. 

239.  Money  is  the  measure  of  value. 

It  is  also  called  Currency,  and  is  of  two  kinds,  viz:   coin 
and  paper  money. 

240.  Coin  or  Specie  is  stamped  pieces  of  metal  having 
a  value  fixed  by  law. 

241.  Paper  Money  is  notes  and  bills  issued  by  the 
Government  and  banks,  and  authorized  to  be  used  as  money. 

UNITED  STATES  MONEY. 

242.  The  unit  of  United  States  or  Federal  money  is  the 
Dollar. 

TABLE. 

10  Mills  (m.)  =  1  Cent    .  .  .  ct. 

10  Cents  =  1  Dime  .  .  .  d. 

10  Dimes  =  1  Dollar.  .  .  $ 

10  Dollars  =  1  Eagle  .  .  .  E. 

$       d.        ct.         m. 
1  =  10  =  100  =  1000 
Scale  —  Decimal. 

The  coins  of  the  United  States  are  — 


:  The  double-eagle,  eagle,  half-eagle,  quarter-eagle,  three- 
dollar  piece,  one-dollar  piece. 

Silver:  The  dollar,  half-dollar,  quarter-dollar,  the  twenty-cent 
piece,  the  ten-cent  piece. 

Nic  kel  :    The  five-cent  piece  and  three-cent  piece. 

Bronze.:    The  one-cent  piece. 

There  are  various  other  coins  of  the  United  States  in  circulation, 
but  they  are  not  coined  now. 


MEASUEES    OF  VALUE.  159 

The  denominations  dimes  and  eagles  are  rarely  used,  the  dimes 
being  regarded  as  cents,  and  the  eagles  as  dollars. 

No  examples  in  Reduction  of  U.  S.  Money  are  given,  because  the 
pupil  has  been  familiarized  with  the  process  from  the  beginning. 

CANADA  MONEY. 

243.  The  currency  of  Canada  is  decimal,  and  the  table  and 
denominations  are  the  same  as  those  of  United  States  money. 
English  money  is  still  used  to  some  extent. 

The  coins  of  Canada,  are,  for  the  most  part,  of  the  same  denomina- 
tions as  those  of  the  United  States,  except  the  gold  coins,  which  are 
the  sovereign  and  half-sovereign. 

ENGLISH  OR  STERLING  MONEY. 

244.  English  money  is  the  currency  of  Great  Britain.    The 
unit  is  the  Pound  or  Sovereign. 

TABLE. 

4  Farthings  (far.)     =     1  Penny.     .     .    d. 
12  Pence  =     1  Shilling  .     .     s. 

OA  01  «iv  f  1  Pound,  or)        „ 

20  Shillings  =  1  .  !      >  •    £ 

( 1  Sovereign  J 

£       s.         d.        far. 
1  =  20  =  240  =  960 
Scale  — 4,  12,  20. 

1.  Farthings    are   commonly   written    as    fractions   of    a    penny. 
Thus,  7  pence  3  farthings  is  written  7|d.;  5  pence  1  farthing,  5^d. 

2.  The  value  of  £1  or  sovereign  is  $4.8665  in  American  gold. 

The  coins  of  Great  Britain  in  general  use  are — 

Gold:  Sovereign,  half-sovereign,  and  guinea,  which  is  equal  to 
21  shillings. 

Silver  :  The  crown  (equal  to  5  shillings),  half-crown,  florin  (equal 
to  2  shillings),  shilling,  six-penny  and  three-penny  pieces. 

Copper:   Penny,  half-penny,  and  farthing. 


160  DENOMINATE   NUMBERS. 


REDUCTION  DESCENDING. 

245.  1.  How  many  farthings  are  there  in  2  pence?     In  5 
pence?     In  7  pence?     In  8  pence?     In  6  pence? 

2.  How  many  pence  are  there  in  2  shillings?     In  5  shil- 
lings?    In  7  shillings?     In  8  shillings?     In  6  shillings? 

3.  How  many  pence  are  there  in  5s.  ?    In  5s.  and  3d.  ?    In 
7s.  4d.?     In  4s.  5d.?     In  6s.  8d.? 

4.  How  many  farthings  are  there  in  5d.  ?    In  6d.  3  far.  ?    In 
5id.?     In6id.?     InSfd.?     InG^d.?     InlOfd.? 

5.  How  many  shillings  are  there  in  £2  5s.  ?     In  £3  5s.  ? 

246.  Reduction  of  a  denominate  number  is  the  process 
of  changing  it  from  one  denomination  to  another  without 
altering  its  value. 

247.  Reduction  Descending  is  the  process  of  chang- 
ing a  denominate  number  to  an  equivalent  number  of  a  lower 
denomination. 

WRITTEN    EXERCISES. 

1.  How  many  farthings  are  there  in  £3  5s.  6f  d.  ? 

PROCESS.  ANALYSIS. — Since    in    1    pound 

-£S   ^s    6^-d  there  are  20  shillings,  in  3  pounds 

^  ^  there  are  3  times  20  shillings,  or  60 

shillings;  and  60  shillings  +  5  shil- 
65s.        =  £  3  5  S.  lings  =  65  shillings. 

1  2  Since  in  1  shilling  there  are  12 


786d       — £35s    6d  pence,  in  65  shillings  there  are  65 

^  times  12  pence,  or  780  pence;  and 

780  pence  +  6  pence  =  786  pence. 


3  1 47  far.  =  £3  5s.  6|d.  Since  in  1  penny  there  are  4  far- 

things, in  786  pence  there  are  3144 

farthings;  and  3144  farthings  +  3  farthings  =  3147  farthings. 
Hence  in  £3  5s.  6|d.  there  are  3147  farthings. 


REDUCTION    DESCENDING.  161 

2.  How  many  pence  are  there  in  £2  10s.  6d.  ? 

3.  How  many  shillings  are  there  in  £13  5s.  ? 

4.  How  many  farthings  are  there  in  £4  6s.  5d.  ? 

5.  How  many  pence  are  there  in  £|-? 

PROCESS.  ANALYSIS. — Since  in  1  pound 

£  3  _ -   3.   of  20  s  =  -6-°-S  there  are  20  shillings,  in  f  of  a 

pound  there   are  f  of  20  shil- 
6/s.  =  Sf.  of  12 d.  =  -^-pd.        iingSj  or  _67o  of  a  shilling. 

720  j  __  102^d.  Since  in  1  shilling  there  are 

12  pence,   in  sf-  of   a  shilling 

there  are  -6^  of  12  pence,  or  -S-f9-  pence;  and  -^f2-  pence  =  102fd. 
Therefore  in  £f  there  are  102fd. 

EULE. — Multiply  the  number  of  the  highest  denomination  given, 
by  the  number  of  units  of  the  next  lower  denomination  which  is 
equal  to  one  of  the  next  higher,  and  to  the  product  add  the  num- 
ber given  of  this  lower  denomination. 

Proceed  in  like  manner  with  this  and  each  successive  result 
thus  obtained,  until  the  number  is  reduced  to  the  denomination 
required. 

6.  How  many  pence  are  there  in  £f  ? 

7.  How  many  pence  are  there  in  ££? 

8.  How  many  farthings  are  there  in  -f s.  ? 

9.  How  many  pence  are  there  in  £T5T? 

10.  How  many  shillings  are  there  in  £5  6s.?     How  many 
farthings  ? 

11.  Keduce  12s.  5d.  2  far.  to  farthings. 

12.  How  many  pence  are  there  in  £7  9s.  5d.  ? 

13.  Eeduce  17s.  6fd.  to  farthings. 

14.  What  is  the  value  of  £f  jn  units  of  lower  denomina- 
tions? 

15.  Find  the  number  of  farthings  in  £5  13s.  3d. 

16.  Reduce  £35  6s.  8d.  to  pence. 

17.  Reduce  £45  3s.  9|d.  to  farthings. 

18.  Reduce  £29  18s.  5d.  to  farthings. 

11 


162  DENOMINATE    NUMBERS. 


REDUCTION  ASCENDING. 

248.  1.  How  many  pence  are  there  in  12  farthings?  In 
16  farthings?  In  20  farthings? 

2.  How  many   shillings  are   there  in   24  pence?     In  60 
pence?     84  pence?     96  pence? 

3.  How  many  pounds  are  there  in  40  shillings?     In  60 
shillings?     In  120  shillings? 

4.  How  many  pounds  sterling  must  be  paid  for  10  pairs 
of  boots  at  6  shillings  a  pair? 

5.  At  5  shillings  each  how  many  pounds  sterling  must  be 
paid  for  16  hats?     For  20  hats? 

6.  Sold  8  pairs  of  skates  at  5  shillings  a  pair.     How  many 
pounds  sterling  did  I  receive  for  them? 

Reduction  Ascending  is  the  process  of  changing  a 
denominate  number  to  an  equivalent  number  of  a  higher 
denomination. 

WRITTEN    EXERCISES. 

1.  How  many  pounds  sterling  are  there  in  7254  pence? 

PROCESS.  ANALYSIS.  —  Since  12  pence  are 

12)7254  equal  to  1   shilling,   there  must 


be  as  many  shillinSs    in 
pence  as  12  pence  are  contained 

30  ....  4  times  in  that  number.     12  pence 

„  n  r~  A  i          ^  o  A  n  i         are  contained  in  7254  pence  604 

7254  d.  =£30    4s.  6d.       A.         .Al  .   ,     *7- 

times  with  a  remainder  ot  6  pence, 

therefore  7254  pence  are  equal  to  604s.  6d. 

Since  20  shillings  are  equal  to  1  pound,  there  must  be  as  many 
pounds  in  604  shillings  as  20  shillings  are  contained  times  in  that 
number.  20  shillings  are  contained  in  604  shillings  30  times  and  a 
remainder  of  4  shillings. 

Therefore  7254  pence  are  equal  to  £30  4s.  6d. 


REDUCTION    ASCENDING.  163 

2.  How  many  shillings  are  there  in  345  farthings? 

3.  How  many  pounds  are  there  in  456  shillings? 

4.  How  many  pounds  are  there  in  1586  pence? 

5.  Reduce  3864  farthings  to  pounds. 

6.  Reduce  f  d.  to  a  fraction  of  a  pound. 

PROCESS.  ANALYSIS. — Since   1    penny  is 

3  rl   -      3    rvf    i   Q          3  «  A  of  a  shilling,  f  of  a  penny  is 

1  T2  s-    -  -8*s-          equai  to  f  of  ^  of  a  shilling,  or 
/¥S.  =  &  Of  £3^  ==  £T^      A  of  a  shilling.  _ 

Since   1    shilling    is    -fa   of    a 

pound,  -fx  of  a  shilling  is  equal  to  y\  of  YV  °f  a  pound,  or  Tg3^  of  a 
pound. 

RULE. — Divide  the  given  number  by  the  number  of  that  de- 
nomination which  is  equal  to  a  unit  of  the  next  higher  denomi- 
nation. 

Divide  the  quotient  in  like  manner,  and  thus  proceed  until  the 
required  denomination  is  reached. 

The  last  quotient  and  the  several  remainders  ivill  be  the  result 
sought. 

7.  Change  f  of  a  shilling  to  a  fraction  of  a  pound. 

8.  Change  -f-  of  a  farthing  to  a  fraction  of  a  shilling. 

9.  Change  384  pence  to  units  of  higher  denominations. 
10.  Change  3146  shillings  to  pounds. 


Reduce : 

11.  3596d.  to  pounds. 


Reduce  : 

20.  £15  8s.  to  farthings. 


12.  3846  far.  to  shillings.          21.  £15  to  dollars. 

13.  4856s.  to  pounds.  22.  $456  to  pounds. 


14.  5968  far.  to  pounds. 

15.  3984d.  to  pounds. 

16.  4685  far.  to  shillings. 

17.  48567  far.  to  pounds. 

18.  £3  14s.  5d.  to  far. 

19.  48596  far.  to  pounds. 


23.  $394.45  to  pounds. 

24.  $37.50  to  pounds. 

25.  £25  to  dollars. 

26.  £15  10s.  to  farthings. 

27.  $973.30  to  pounds. 

28.  $1216.625  to  pounds. 


164  DENOMINATE    NUMBEKS. 


FRENCH  MONEY. 

249.  In  France  the  currency  is  decimal.     The  unit  is  the 

Franc. 

TABLE. 

10  Centimes  (ct.)  [pronounced  son-teems]  =  1  Decime     .     .    dc. 
10  Decimes  [pronounced  des-seems]  =  1  Franc  .     .     .     fr. 

Scale  —  Decimal. 

The  value  of  the  franc,  as  determined  by  the  Secretary  of  the  Treas- 
ury, is  $  .193  in  United  States  money. 

1.  How  many  centimes  are  there  in  1  franc?     In  5  francs? 

2.  How  many  decimes  are  there  in  1  franc  ?     In  7  francs  ? 

3.  How  many  centimes  are  there  in  4  decimes? 

4.  How  many  dollars  are  there  in  10  francs?  In  20  francs? 

5.  In  3684  centimes  how  many  francs  are  there? 

6.  How  many  francs  are  there  in  $19.30?     In  $9.65?    In 
$3.86? 

MEASURES  OF  SPACE. 

250.  Space  is  extension  in  any  direction.     It  has  three 
dimensions  or  measurements — length,  breadth  and  thickness. 

251.  A  Line  is  that  which  has  only  length. 

Thus,  the  edge  of  any  thing,  or  the  distance  between  any  two  objects 
or  places,  is  a  line. 

252.  A  Surface   is   that   which   has    only  length   and 
breadth. 

Thus,  the  floor,  this  page,  or  the  outside  of  any  thing,  is  a  surface. 

253.  A  Solid  is  that  which    has  length,,  breadth   and 
thickness. 

Thus,  a  stone,  an  apple,  a  block,  a  book,  etc.,  are  solids. 


MEASURES   OF   SPACE. 


165 


LINEAR  MEASURES. 

254,  Linear  Measures  are  used  in  measuring  lengths 
and  distances. 


SUEVEYOES'  LINEAE  MEAS. 


LINEAE 

MEASUEE. 

12  Inches  (in.) 

=  1  Foot    . 

ft. 

3  Feet 

=  1  Yard   . 

yd. 

5J  Yards| 
16£  Feet    J 

=  1  Eod     . 

rd. 

320  Eods 

=  1  Mile    . 

mi. 

mi.     rd. 

yd. 

7.92  Inches    —  1  Link  . 

25  Links     =  1  Eod    . 
4  Eods 
100  Links 

80  Chains   =  1  Mile 

ft.  in. 


1  Chain 


1. 
rd. 

ch. 
mi. 


Scale - 


1  =  320  =  1760  =  5280  =  63360 
-12,  3,  51   and  320. 


The  following  are  also  used: 


3  Barleycorns 

4  Inches 
6  Feet 

3  Feet 

5  Paces 

8  Furlongs 
1.15  Statute  Miles 
3  Geographical  Miles 
60  Geographic  Miles  ^) 
69.16  Statute  Miles   f 


Used  in  pacing  distances. 


:  1  Inch.    Used  by  shoemakers. 
:  1  Hand.   Used  to  measure  the  height  of  horses. 
:  1  Fathom.     Used  to  measure  depths  at  sea. 
=  1  Pace.) 
=  1  Eod.  J 
:1  Mile. 

=  1  Geographical,  or  Nautical  Mile. 
League. 

fof  Latitude  on  a  Meridian,  or 
'  Longitude  on  the  Equator. 


=  lDegree{j 


1.  For  the  purpose  of  measuring  cloth  and  other  goods  sold  by  the 
yard,  the  yard  is  divided  into  halves,  fourths,  eighths,  and  sixteenths. 

2.  The  length  of  a  degree  of  latitude  varies.     69.16  is  the  average 
length,  and  is  that  adopted  by  the  United  States  Coast  Survey. 


166  DENOMINATE    NUMBEKS. 

1.  How  many  inches  are  there  in   4  feet?     6  feet?  ^8 
feet?     10  feet?     12  feet? 

2.  How  many  feet  are  there  in  2  rods?     3  rods?     4  rods? 

3.  How  many  inches  are  there  in  2  yards?     4  yards?     5 
yards  ? 

4.  How  many  inches  are  there  in  2  yards  and  2  inches? 
3  yards  and  4  inches? 

5.  How  many  rods  are  there  in  2  miles?     3  miles? 

6.  How  many  feet  are  there  in  1  rod  and  2  yards?     2 
rods  and  3  yards? 

7.  How  many  feet  are  there  in  45  inches?     In  63  inches? 

8.  How  many  yards  are  in  22  feet?     In  47  feet?     In  34 
feet? 

9.  How  many  miles  in  640  rods?     In  480  rods? 

10.  How  many  inches  in  10  links?     In  100  links? 

11.  How  many  links  in  5  rods?     In  3  rods?     In  6  rods? 

12.  The  length  of  a  road  was  400  links.   What  was  its  length 
in  rods? 

13.  In  160  chains  how  many  miles? 

WRITTEN    EXERCISES. 

14.  Reduce  5  mi.  18  rd.  4  yd.  to  yards. 

15.  Reduce  7  rd.  5  ft.  6  in.  to  inches. 

16.  How  many  inches  are  there  in  7  miles?     In  9  miles? 

17.  A  building  was  327  ft.  long.     How  many  rods  was  it 
in  length? 

18.  A  man  sold  a  piece  of  wire  36828  in.  long.     How 
many  rods  was  it  in  length? 

19.  In  3960  rods  how  many  miles  are  there? 

20.  Reduce  15  mi.  8  rd.  5  yd.  3  ft.  4  in.  to  inches, 

21.  Reduce  8  mi.  14  rd.  5  ft.  4  in.  to  inches. 

22.  Reduce  66454  inches  to  miles,  etc. 

23.  Reduce  158964  inches  to  miles,  etc. 


MEASUKES   OF  SPACE. 


167 


24.  The  diameter  of  the  earth  is  7912  miles.     How  many 
feet  is  it? 

25.  How  high  is  a  horse  that  measures  15  hands? 

26.  My  farm  is  67  ch.  83  1.  long.     How  many  rods  long 
is  it? 

27.  Keduce  59  ch.  75  1.  to  inches. 


: 


SURFACE  MEASURES. 


255.  An  Angle  is  the  difference  in 
the  direction  of  two  lines  that  meet. 

256.  A   Square   is  a   figure  that 
has  four  equal    sides,   and   four   equal 
angles. 

A  square  inch  is  a  square  whose  side  is  one 
inch.  A  square  foot,  a  square  whose  side  is 
one  foot. 

The  angles  of  a  square  are  called  right  angles. 

257.  A  Rectangle  is  a  figure  that 
has  four  straight  sides  and  four  equal 
angles. 

The  angles  of  a  rectangle  are  all  right  angles. 

258.  The   Area   or    extent  of  any 
surface  is  the  number  of  square  units  it 
contains. 

Thus,  if  a  rectangle  is  4  inches  long  and 
3  inches  wide  the  area  will  be  12  square 
inches. 

For  it  may  be  divided  into  4  rows,  each 
containing  3  square  inches  or  units,  and  the 
entire  area  will  be  12  square  inches. 

The  method  of  computing  the  area  of  fig- 
ures that  are  not  rectangular  is  given  in 
MENSURATION. 


ANGLE. 


SQUARE. 


KECTAJSGLE. 


168  DENOMINATE   NUMBERS. 

259.   The  area  of  a  rectangle  is  equal  to  the  product  of  the 
numbers  that  express  its  length  and  breadth. 

The  length  and  breadth  must  be  expressed  in  units  of  the  same  de- 
nomination. 

1.  How  many  square  inches   are   there   in  p,  rectangle  6 
inches  long  and  5  inches  wide?     8  inches  long  and  3  inches 
wide?     7  inches  long  and  5  inches  wide? 

2.  How  many  square  feet  are  there  in  a  rectangle  4  feet 
long  and  3  feet  wide?     7  feet  long  and  5  feet  wide? 

3.  How  many  square  inches  are  there  in  a  square  whose 
side  is  2  inches?     5  inches?     8  inches?     12  inches? 

4.  How  many  square  yards  are  there  in  a  square  whose 
side  is  2  yards?     5  yards?     7  yards?     10  yards? 

5.  How  many  square  feet  are  there  in  a  square  whose  side 
is  1  yard  long?     3  yards?     5  yards?     7  yards?     10  yards? 

6.  How  many  square  rods  are  there  in  a  lot  5  rods  long 
and  4  rods  broad?     In  a  square  whose  side  is  6  rods? 

7.  How  many  square  feet  in  a  square  whose  side  is  3  yards? 
In  a  rectangle  whose  length  is  4  yards  and  breadth  3  yards? 

8.  How  many  square  inches  are  there  in  a  square  foot? 
Square  feet  in  a  square  yard?     Square  yards  in  a  square  rod? 

SQUARE  MEASURE. 

TABLE. 

144  Square  Inches  (sq.  in.)  =  1  Square  Foot    .     .     .  sq.  ft. 

9  Square  Feet  =  1  Square  Yard  .     .     .  sq.  yd. 

30|  Square  Yards  =  1  Square  Hod     .     .     .  sq.  rd. 

160  Square  Rods  =  1  Acre       A. 

640  Acres  =  1  Square  Mile    .     .     .  sq.  mi. 

sq.mi.  A.         sq.rd.         sg.  yd.  sq.ft.  sq.in. 

1  =  640  =  102400  =  3097600  =  27878400  =  4014489600 

Scale— 144,  9,  30J,  160,  640. 


MEASUKES    OF   SPACE.  169 

1.  Plastering,  ceiling,  etc.,  are  commonly  estimated  by  the  square 
yard;  paving,  glazing,  and  stone-cutting,  by  the  square  foot. 

2.  Hoofing,  flooring   and.  slating   are  commonly  estimated  by  the 
square  of  100  feet. 

SURVEYORS'  SQUARE  MEASURE. 


TABLE. 


625  Square  Links    =  1  sq.  rd. 
16  Square  Kods    —  1  sq.  chain. 


10  Square  Chains       =1  acre. 
640  Acres  —  1  sq.  mi. 


In  some  parts  of  the  country  a  Township  contains  36  square  miles, 
or  is  6  miles  square. 

1.  How  many  square  feet  are  there  in  4  square  yards? 
7  square  yards?     9  square  yards? 

2.  How  many  square  inches  are  there  in  2  square  feet? 
3  square  feet?     5  square  feet? 

3.  How  many  square  yards  are  there  in  27  square  feet? 
36  square  feet?     81  square  feet? 

4.  How  many  square  yards  are  there  in  10  square  rods? 

5.  How  many  square  chains  are  there  in  48  square  rods  ? 
64  square  rods?     96  square  rods? 

6.  How  many  square  rods  are  there  in  3  acres?     In  5 
acres? 

7.  How  many  acres  are  there  in  480  square  rods? 

8.  How  many  square  feet  are  there  in  288  square  inches? 

9.  How  many  acres  are  there  in  30  square  chains?     In 
50? 

WRITTEN     EXERCISES. 

10.  Reduce  9  sq.  yd.  3  sq.  ft.  15  sq.  in.  to  square  inches. 

11.  Reduce  3  sq.  mi.  15  sq.  rd.  to  square  inches. 

12.  Reduce  262685  sq.  ft.  to  acres,  etc. 

13.  Reduce  2  A.  37  sq.  rd.  5  sq.  yd.  7  sq.  ft.  to  sq.  in. 

14.  Reduce  184265  sq.  in.  to  units  of  higher  denominations. 


170  DENOMINATE   NUMBERS. 

15.  Reduce  -f-  of  an  acre  to  units  of  lower  denominations. 

PROCESS. 

f    A.  X  1 60  r=  -»00  gq.  r(J.  —  H42  gq>  r(J. 

f  sq.  rd.  X  30 1  =  f  sq.  rd.  X  H1  =  W  sq«  7d- 
&sq.yd.X      9  =  H  sq.ft.  =  5^  sq.ft. 

iJ sq.ft.    X144  =  iff4sq.m.       =  H3^Bq.in. 

Therefore  f  A.  — 114  sq.  rd.  8  sq.  yd.  5  sq.  ft.  113f  sq.  in. 

ANALYSIS. — We  multiply  by  that  number  in  the  scale  which  will 
reduce  the  number  to  the  next  lower  denomination,  and  so  continue 
to  multiply  each  fraction  until  the  lowest  denomination  is  reached. 

16.  Express  f  of  an  acre  in  lower  denominations. 

17.  What  part  of  an  acre  are  100  sq.  rd.?     80  sq.  rd.? 
120  sq.  rd.? 

18.  Change  f  of  a  sq.  rd.  to  lower  denominations. 

19.  How  many  sq.  in.  are  there  in  a  rectangle  7  inches 
wide  by  11  inches  long? 

20.  How  many  square  feet  are  there  in  a  floor  8  feet  long 
by  15  feet  wide? 

21.  How  many  square  yards  are  there  in  a  ceiling  that  is 
18  feet  wide  by  21  feet  long? 

22.  What  is  the  area  of  a  square  whose  side  is  5  feet? 

23.  How  many  square  yards  are  there  in  a  floor  18  feet 
wide  by  24  feet  long?     How  much  would  it  cost  to  carpet 
it  at  $1.15  per  square  yard? 

24.  How  many  yards  of  carpeting   1  yard  wide  will  be 
required  to  cover  a  room  18  ft.  long  by  17  ft.  wide? 

25.  What  will  it  cost  to  carpet  a  room  18  ft.  long  by  15f 
ft.  wide,  with  carpet  f  of  a  yard  wide,  at  $1.90  per  yard? 

26.  If  the  width  of  a  lot  is  66  feet,  how  long  must  it  be 
to  contain  \  of  an  acre?     What  will  be  the  cost  of  it  at 
$3.25  per  square  foot? 

27.  A  pasture  containing  10  acres  had  a  width  of  20  rods? 
How  long  was  it? 


MEASURES   OF  VOLUME. 


171 


28.  Mr.  A.  sold  a  lot  of  land  whose  width  was  20  rd. 
and  whose  length  was  80  rd.   at  $47.25   per  acre.     How 
much  did  he  get  for  it? 

29.  What  is  the  difference  between  10  square  feet  and  10 
feet  square?     Illustrate  this  by  drawings. 

30.  What  will  be  the  expense  of  painting  a  roof  48  feet 
long  and  22  feet  wide  at  $.30  a  square  yard? 

31.  What  will  be  the  cost  of  cementing  the  bottom  of  a 
cellar  45  feet  by  32  feet  at  $.30  per  square  yard? 

32.  How  many  yards  of  plastering  are  there  in  the  sides 
of  a  room  18  ft.  long,  17  ft.  wide,  and  11  ft.  high?     How 
many  in  the  ceiling  ?     What  will  be  the  cost  of  plastering  at 
$.37  a  square  yard? 

33.  What  will  be  the  cost  of  papering  the  side  walls  of 
the  above  room  at  $.25  per  square  yard? 


MEASURES  OF  VOLUME. 

260.  A  Solid  has  length,  breadth,  and  thickness. 

261.  A  Cube  is  a  solid  having 
six  equal  square  sides  called  faces. 

262.  A  Cubic  Inch  is   a 

solid  whose  sides  or  faces  are  each 
a  square  inch. 

263.  A  Cubic  Foot  is  a  solid 
whose  sides  are  each  a  square  foot. 

264.  The  Volume,  or  Solid 

Contents,  of  any  body  is  the 
number  of  solid  units  it  contains. 

Thus,  if  a  solid  is  4  ft.  long,  3  ft. 
wide,  and  3  ft.  thick,  its  volume  will 


172 


DENOMINATE    NUMBERS. 


be  36  cubic  feet.  For  it  may  be  divided  into  3  blocks,  each  contain- 
ing 12  cubic  feet,  making  in  all  36  cubic  feet.  That  is,  the  number 
of  cubic  feet  in  each  block  will  be  equal  to  the  product  of  the  num- 
bers expressing  its  length  and  breadth,  and  the  number  of  blocks  is 
equal  to  the  number  expressing  the  thickness.  Therefore, 

265.   The  volume  of  any  rectangular  solid  is  equal  to  the  prod- 
uct of  the  numbers  expressing  its  length,  breadth,  and  thickness. 

The  length,  breadth  and  thickness  must  be  expressed  in  units  of 
the  same  denomination. 

1.  How  many  cubic  feet  are  there  in  a  rectangular  solid 
whose  length  is  3  ft. ,  its  breadth  2  ft. ,  and  its  thickness  2  ft.  ? 

2.  How  many  cubic  feet  are  there  in  a  cube  whose  dimen- 
sions are  each  3  feet;   or,  how  many  cubic  feet  are  there  in 
a  cubic  yard  ?     In  a  cube  whose  sides  are  5  ft.  long  ? 

3.  How  many  cubic  inches  are  there  in  a  cube  whose  di- 
mensions are  each  12  inches;  or,  how  many  cubic  inches  are 
there  in  a  cubic  foot?     In  a  cube  whose  sides  are  10  in.  long? 

4.  What  is  the  volume  of  a  cube  whose  sides  are  each  4 
inches  square?     9  inches  square?     16  inches  square? 


CUBIC  MEASURE. 

TABLE. 

1728  Cubic  Inches  (cu.  in.)  =  1  Cubic  Foot 
27  Cubic  Feet  =  1  Cubic  Yard 


cu.  ft. 
cu.  yd. 


A  cord  of  wood  or 
stone  is  a  pile  8  feet 
long,  4  feet  wide  and  4 
feet  high. 

A  pile  that  is  1  foot 
long,  4  feet  wide  and  4 
feet  high,  is  a  cord  foot. 


MEASURES   OF   VOLUME.  173 

following  are  the  denominations: 

16  Cubic  Feet    =  1  Cord  Foot     .     .     .    cd.  ft. 
8  Cord    Feet) 


128  Cubic  Feetf" 

1.  A  perch  of  stone  or  masonry  is  16J  ft.  long,  1|  ft.  thick,  and  1  foot 
high,  and  contains  24}  cu.  ft. 

2.  A  cubic  yard  of  earth  is  considered  a  load. 

3.  Brick-work  is  commonly  estimated  by  the  thousand  bricks. 

4.  Brick-layers,  masons  and  joiners  commonly  make  a  deduction 
of  one-half  the  space  occupied  by  windows  and  doors  in  the  walls  of 
buildings. 

5.  In  computing  the  contents  of  walls,  masons  and  brick-layers  mul- 
tiply the  entire  distance  around  on  the  outside  of  the  wall  by  the 
height  and  thickness.    The  corners  are  thus  measured  twice. 


WRITTEN     EXERCISES. 

1.  How  many  cubic  inches  are  there  in  2  cubic  feet?     In 
3  cu.  ft.  ?    In  15  cu.  ft.  ?    In  32  cu.  ft.  ? 

2.  How  many  cubic  feet  are  there  in  2  cubic  yards?     In 
3  cu.  yd.  ?     In  13  cu.  yd.  ?     In  25  cu.  yd.  ? 

3.  How  many  cubic  feet  are  there  in  5  cords?    In  8  cords? 

4.  How  many  perch  of  masonry  are  there  in  418  cubic  feet? 
What  will  be  the  cost  of  laying  it  at  $1.75  per  perch? 

5.  How  many  perch  of  masonry  are  there  in  a  wall  38  feet 
long,  4  feet  high,  and  1^  feet  thick? 

6.  How  many  yards  or  loads,  of  earth,  must  be  removed 
in  digging  a  cellar  35  feet  by  20,  8  feet  deep? 

7.  Keduce  32  cu.  ft.  114  cu.  in.  to  cubic  inches. 

8.  Keduce  13  cu.  yd.  18  cu.  ft.  to  cubic  feet. 

9.  Reduce  15  perch  13^-  cu.  ft.  to  cubic  feet. 

10.  How  many  cubic  blocks  of  one  foot  on  a  side  can  be 
cut  from  a  cube  that  is  8  yards  long  on  each  edge? 

11.  How  many  cubic  feet  in  a  block  of  marble  9  feet  long, 
5  feet  wide,  and  3£  feet  thick  ? 


174 


DENOMINATE   NUMBEKS. 


12.  A  man  sawed  a  pile  of  wood  40  ft.  long,  4  ft.  wide,  and 
5|  ft.  high,  for  $1.50  per  cord.     How  much  did  he  earn? 

13.  A  bin  is  8  ft.  long,  7  ft.  wide,  and  5  ft.  high.     How 
many  cubic  feet  are  there  in  it?     How  many  cubic  inches? 
How  many  bushels  will  it  hold  if  a  bushel  contains  2150.4 
cubic  inches? 

14.  What  will  it  cost  to  excavate  a  cellar  80  by  35  ft. ,  and 
8  ft.  deep,  at  $  .42  per  yd.  ?    What  will  be  the  expense  of  build- 
ing a  stone  wall  around  it  1^  ft.  thick,  at  $3.75  a  perch? 

15.  How  many  bricks  will  it  require  to  build  a  wall  35^- 
ft.  long,  19  ft.  high,  and  3  ft.  thick,  allowing  22  bricks  to 
the  cubic  foot  when  laid  ? 


BOARD  MEASURE. 

266.  In  measuring  lumber,  when  a  board  is  one  inch  thick, 
the  number  of  feet  board  measure  is  obtained  by  multiplying 
the  length  in  feet  by  the  breadth  expressed  in  feet. 

When  the  lumber  is  more  than  one  inch  thick,  the  number 
of  feet,  board  measure,  may  be  obtained  by  multiplying  the 
length  in  feet  by  the  breadth  in  feet,  and  this  product  by  the 
number  expressing  the  inches  in  thickness. 

When  a  board  tapers  uniformly, 
the  average  or  mean  width  is 
equal  to  half  the  sum  of  the  two 
ends. 


Board  measure  may  also  be  computed  by  multiplying  the  number 
of  feet  in  length  by  the  number  of  inches  in  width,  and  then  dividing 
the  product  by  12. 


EXERCISES. 


How  many  feet  are  there  in  the  following  boards: 


1.  18  ft.  by  16  in.? 

2.  15  ft.  by  11  in.? 


3.  10  ft.  by  13  in.? 

4.  13  ft.  by  15  in.  ? 


MEASURES  OF  CAPACITY.  175 

5.  How  many  feet  of  timber  are  there  in  a  stick  40  feet 
tig,  9  inches  wide,  and  6  inches  thick? 

6.  Mr.  B.  bought  318  fence  boards  16  feet  long  and  8  inches 
vide.     What  did  they  cost  at  $11  per  thousand  feet? 

7.  A  lumber  dealer  bought  35  three-inch  planks,  22  feet 
long  and  16  inches  wide,  at  $17.50  per  M.     How  much  did 
they  cost? 

8.  What  will  it  cost  to  floor  a  room  35  feet  by  18,  with  1^ 
inch  flooring,  at  $30  per  M,  allowing  ^  for  matching? 

9.  What  will  be  the  expense  of  flooring  a  room  20  feet 
by  25  with  1|  inch  flooring,  at  $25  per  M,  allowing  -|-  for 
matching  ? 


MEASURES   OF  CAPACITY. 

LIQUID  MEASURE. 
267.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE. 

4  Gills  (gi.)  =  1  Pint  .  .  .  pt. 
2  Pints  =1  Quart  .  .  .  qt. 
4  Quarts  =1  Gallon  .  .  .  gal. 

gal.      qt.      pt.       gL 
I  =  4  •=  8  =  32 
Scale— 4,  2,  4. 

1.  In  determining  the  capacity  of  cisterns,  reservoirs,  etc.,  31  \  gallons 
are  considered  a  barrel  (bbl.),  and  2  barrels,  or  63  gallons  a  hogshead 
(hhd.).    In  commerce,  however,  the  barrel  and  hogshead  are  not  fixed 
measures. 

2.  Cash  of  large  size  do  not  hold  any  fixed  quantity.    Their  ca- 
pacity is  usually  marked  upon  them. 

3.  The  standard  gallon  of  the  United  States  contains  231   cubic 
inches. 

4.  The  beer  gallon  is  not  now  in  use.     It  contained  282  cubic  inches. 


176  DENOMINATE   NUMBERS. 


EX  EXCISES. 

1.  How  many  gills  are  there  in  3  pints?   5  pints?   7  pints? 

2.  How  many  gills  are  there  in  2  quarts?     3  quarts? 

3.  How  many  pints  are  there  in  3  quarts?     8  quarts? 

4.  How  many  pints  are  there  in  a  cask  which  contains 
37  gallons? 

5.  A  man  sold  684  pints  of  milk  at  20  cents  a  gallon. 
How  much  did  he  get  for  it?    How  many  gallons  were  there? 

6.  Eeduce  3846  gi.  to  gal.     4869  pt.  to  gal. 

7.  Reduce  3  gal.  4  qt.  1  pt.  3  gi.  to  gi. 

8.  Reduce  4  bbl.  6  gal.  to  gi.     484  pt.  to  gal. 

9.  Reduce  24  gal.  to  pt.     8459  gi.  to  bbl. 

10.  How  many  cubic  inches  are  there  in  7  gal.  ? 

11.  How  many  gallons   will  a  vessel   hold   that  contains 
3846  cubic  inches? 

12.  How  many  barrels  of  water  will  a  cistern  hold  that  is 
15  feet  long,  10  feet  wide,  and  8  feet  deep? 


APOTHECARIES'  LIQUID  MEASURE. 

268.  Apothecaries9  Liquid  Measure  is  used  in 
compounding  and  measuring  liquid  medicines. 

TABLE. 

60  Drops  (gtt.)  or  minims  (Til)  — 1  Fluid  drachm  .  /£. 

8  Fluid  drachms  =  1  Fluid  ounce     .  /§. 

16  Fluid  ounces  =lPint         ...  0. 

8  Pints  =  1  Gallon     .     .     .  Cong. 

1.  The  abbreviation  Cong,  is  from  the  Latin  congim,  a  gallon.     A 
pint  being  one-eighth  of  a  gallon  the  abbreviation  is  0.,  from  the  Latin 
octavus,  one-eighth. 

2.  In  writing  prescriptions,  physicians  write  the  number  after  the 
symbol;  thus:  0.  5,  /§  2,  etc. 


MEASURES    OF    CAPACITY.  177 


DRY  MEASURE. 

269.  Dry  Measure  is  used  in  measuring  grain,  roots, 

fruit,  etc. 

TABLE. 

2  Pints  (pt.)  —  1  Quart  .  .  .  qt. 
8  Quarts  =  1  Peck  .  .  .  pk. 
4  Pecks  =  1  Bushel  .  .  .  bu. 

bu.     pk.       qt.        pt. 
1  =  4  =  32  =  64 
Scale  —  2,  8,  4. 

1.  In  measuring  grain,  seeds,  or  small  fruits,  the  measure  must  be 
even  full  or  stricken.    In  measuring  large  fruits,  coarse  vegetables,  corn 
in  the  ear,  etc.,  the  measure  should  be  heaped  at  least  six  inches. 

2.  Five  stricken  bushels  are  considered  equal  to  4  heaped  bushels. 

3.  A  standard  bushel  contains  2150.4  cubic  inches. 

4.  A  pint,  quart,  or  gallon,  dry  measure,  is  more  than  the  same 
quantity  liquid  measure,  for  a  quart  is  ^  of  a  bushel,  or  ^  of  2150.4 
cubic  inches,  which  is  about  67 J  cubic  inches,  while  a  quart  liquid 
measure  is  \  of  231  cubic  inches,  or  57f  cubic  inches. 

Cn.  In.  in  Cn.  In. in          Cii.In.in          Cii.In.in 

One  Gal.  OneQt.  One  Pt.  OneGi. 

Liquid  Meets.    231  57|  28J  7& 

DryMeas.         268f  67J  33f  Sf 


EXERCISES. 

1.  How  many  pints  are  there  in  3  quarts?     7  quarts? 

2.  How  many  quarts  are   there   in  2   pecks?      3   pecks? 
5  pecks?     7  pecks? 

3.  How  many  pints  are  there  in  1   bushel?      3  bushels? 
5  bushels?     8  bushels? 

4.  How  many  pints  are  there  in  3  bu.  3  pk.  5  qt.  1  pt.  ? 

5.  How  many  pints  are  there  in  8  bu.  5  qt.  3  pt.  ? 

6.  Change  16845  qt.  to  units  of  higher  denominations. 

12 


178  DENOMINATE    NUMBERS. 

7.  Change  13965  pt.  to  units  of  higher  denominations. 

8.  Change  57364  qt.  to  units  of  higher  denominations. 

9.  Change  35  bu.  3  pk.  6  qt.  1  pt.  to  pints. 

10.  How  many  cubic  inches  are  there  in  7  bu.  ?     8  bu.  ? 
10  bu.  ?     20  bu.  ? 

11.  How  many  bushels  are  there  in  13846  cu.  in.?     35769 
cu.  in.  ?     48695  cu.  in. 

12.  How  many  cubic  inches  are  there  in  a  bin  8  ft.  long, 
7  ft.  wide,  and  5  ft.  high  ?     How  many  bushels  will  it  hold  ? 

13.  How  many  bushels  will  a  bill  hold  that  is  9  ft.  long, 
6ft.  wide,  and  6  ft.  high? 


MEASURES  OF  WEIGHT. 

270.  Weight  is  the  measure  of  the  force  that  attracts 
bodies  to  the  earth. 

AVOIRDUPOIS   WEIGHT. 

271.  Avoirdupois  Weight  is  used  in  measuring  all 
coarse  and  heavy  articles,  as  hay,  grain,  groceries,  coal,  etc., 
and  the  metals,  except  gold  and  silver. 

TABLE. 

*  16  Ounces  (oz.)         =  1  Pound Ib. 

100  Pounds                 =  1  Hundred- weight     .     cvvt. 
20  Hundred-weight  ==  1  Ton T. 

T.     cwt.       Ib.  oz. 

1  =  20  =  2000  =  32000 

Scale  — 16,  100,20. 

1.  In  weighing  coal  at  the  mines  and  in  levying  duties  at  the  United 
States  Custom  House,  the  lony  ton  of  2240  Ib.  is  sometimes  used. 

2.  The  ounce  is  considered  as  16  drams. 


MEASURES   OF   WEIGHT. 


179 


The  following  denominations  are  also  used: 


56  Ib.  Butter 
100  Ib.  Grain  or  Flour 
100  Ib.  Dried  Fish 
100  Ib.  Nails 
196  Ib.  Flour 
200  Ib.  Pork  or  Beef 


=  1  Firkin. 
=  1  Cental. 
=  1  Quintal. 
=  1  Keg. 
=  1  Barrel. 
=  1  Barrel. 


280  Ib.  Salt  at  N.  Y.  Works  =  I  Barrel. 

The  following  are  the  pounds  in  a  bushel  in  the  States 
named : 


a 

* 

5 

^ 
^' 

^ 

ti 

3 

* 

CO 
V: 

§ 

rC: 
.v) 

^ 

I 

1 

J 

% 

S 
fej 

& 

'§' 

O 

e 

Penna.  \ 

^- 

£ 

1 

C5 

* 

Wheat  

GO 
52 
32 
50 
40 
54 

50 
5C> 
28 

45 

56 

60 
56 

GO 
52 
32 

48 
40 
54 
60 
45 

00 

56 
32 

4S 
50 
56 
60 
45 

GO 
5(5 
32 

48 
52 
56 
60 
45 

60 
56 
33% 

48 
52 
56 
60 
45 

60 
56 
32 
32 

32 

30 

60 
56 
30 
46 
46 
56 

60 
56 
32 
48 
42 
56 
(SO 

GO 
56 
32 

48 
42 
56 
60 

GO 

52 
35 
48 
52 
56 
(JO 
45 

30 

60 
56 
30 
48 
50 
56 
64 

60 
56 

32 

48 
48 
56 
GO 
44 

GO 
56 
32 
48 

56 
60 

60 
56 
34 
46 
42 
56 
60 

60 

56 
32 
47 

48 
56 

60 
56 
32 
46 
4G> 
56 

60 
56 
36 
45 
42 
5G, 
60 

GO 
56 
32 
48 
42 
56 
60 
46 

60 
54 

48 

50 

Indian  Corn 

Oats            

Barley    

Buckwheat  

Rye  

Clover  Seed. 

Timothy  Seed....... 

EXERCISES. 

1.  How  many  ounces  are  there  in  5  Ib.  ?     In  3  Ib.  5  oz.  ? 

2.  How  many  pounds  are  there  in  5  cwt.  ?     In  6  cwt.  ? 

3.  How  many  pounds  are  there  in  1  ton  ?     In  3  T.  ? 

4.  How  many  pounds  are  there  in  3  T.  2  cwt.  5  Ib.  ? 

5.  How  many  pounds  are  there  in  5  T.  216  Ib.  ? 

6-.  How  much  will  5  Ib.  7  oz.   of  indigo  cost  at   $.12|- 
per  oz.  ? 

7.  What  will  3-J-  Ib.  of  confectionery  cost  at  $  .Q4±  per  oz.  ? 

8.  At  8  cents  a  pound,  what  must   be  paid   for  5   cwt. 
28  Ib.  of  sugar? 


180  DENOMINATE    NUMBERS. 

9.  How  many  pounds  are  there  in  \  barrel  of  pork?     In 
\  barrel  of  salt?     In  \  barrel  of  flour?     In  \  keg  of  nails? 

10.  What  will  be  the  value  of  \  barrel  of  flour  at  $8.50 
per  cwt.  ? 

11.  What  will  \  quintal  of  codfish  cost  at  $  .06^  per  lb.? 

12.  What  will  be  the  cost  of  13  cwt.  18  lb.  of  hay  at 
$15  per  ton? 

13.  When  flour  is  $10  a  barrel,  how  many  pounds  can  I 
buy  for  $2.80? 

.    14.  A  merchant  sold  3  cwt.  19  lb.  9  oz.  of  cheese  at  $.17 
per  lb.     How  much  did  he  receive  for  it? 

15.  If  a  merchant  buys  flour  at  $9  per  barrel  and  sells  it 
at  $5  per  cental,  how  much  will  be  his  profit  on  the  sale  of 
15  barrels? 

16.  How  many  barrels  of  salt  are  there  in  275000  lb.  ? 

17.  If  the  weight  of  a  bushel  of  wheat  is  60  lb.,  how  many 
bags  that  hold  2  bu.  each  will  be  required  to  carry  away  3  T. 
4  cwt.  20  lb.  of  wheat? 

TROY  WEIGHT. 

272.  Troy  Weight  is  used  in  weighing  gold,  silver, 

and  jewels. 

TABLE. 

24  Grains  (gr.)     —  1  Pennyweight     .     .     .     pwt. 

20  Pennyweights  =  1  Ounce oz. 

12  ounces  =  1  Pound lb. 

lb.     oz.      pwt.        gr. 
1  =  12  ==240  =  5760 
Scale  —  24,  20,  12. 

1.  In  weighing  diamonds,  pearls,  and  other  jewels,  the  unit  com- 
monly employed  is  the  carat,  which  is  equal  to  4  grains. 

2.  The  term  carat  is  also  used  to  express  the  fineness  of  gold,  and 
means  -fa  part.     Thus,  gold  that  is  18  carats  fine  is  |f  gold  and  £f 
alloy. 


MEASURES   OF   WEIGHT.  181 


APOTHECARIES'  WEIGHT. 

273.  Apothecaries9  Weight  is  used  by  apothecaries 
id  physicians  in  weighing  medicines. 

TABLE. 

20  Grains  (gr.)  =  1  Scruple  .  .  .  sc.,  or  9 

3  Scruples  =  1  Dram  ....  dr.,  or  3 

8  Drams  =  1  Ounce  ....  oz.,  or  § 

12  Ounces          =  1  Pound  ....  lb.,  or  flb 

Ib.      oz.      dr.       sc.         gr. 
1  =  12  =  9Q  =  28S  =  5760 

Scale— 20,  3,  8,  12. 

1.  In  writing  prescriptions,  physicians  express  the  number  in  Ko- 
man  characters,  using  j  instead  of  i  final.     They  also  write  the  symbol 
first;   thus:  §v,  gvj,  ^ij. 

2.  Medicines  are  bought  and  sold  in  large  quantities  by  Avoirdu- 
pois Weight. 

1  lb.  Avoirdupois  =  7000  gr.     1  lb.  JT**/1*1  .    ,  j  =  5760  gr. 

(Apothecaries') 

1  oz.  «  -=  437i  gr.     1  oz.  "  ==  480  gr. 


EXERCISES. 

1.  How  many  grains  are  there  in  3  pwt.  ?     In  5  pwt.  ? 

2.  How  many  pennyweights  are  there  in  5  oz.  ?     In  7  oz.  ? 

3.  How  many  grains  are  there  in  7  oz.  5  pwt.  18  gr.  ? 

4.  Express  3456  grains  Troy  in  higher  units. 

5.  What  will  be  the  value  of  an  ornament  weighing  2  oz. 
15  pwt,  at  $1.35  per  pwt.? 

6.  How  many  spoons,  weighing  5  ounces  each,  can  be  made 
from  3  lb.  5  oz.  of  silver  ? 

7.  How  many  powders,  of  5  grains  each,  can  be  made  from 
5  oz.  7  dr.  of  quinine  ? 


182  DENOMINATE   NUMBERS. 

MEASURES  OF  TIME. 

274.  The  following  are  the  ordinary  divisions  of  time : 
TABLE. 


60  Seconds  (sec.) 

=  1  Minute    .     .     , 

,     min. 

60  Minutes 

=  1  Hour  .     .     .     . 

,     hr. 

24  hours 

=  1  Day     .     .     .     , 

,     da. 

7  days 

=  1  Week  .     .     .     , 

,     wk. 

365  days 

=  1  Year   .     .     .     , 

vr 

366  days 

=  1  Leap  Year   .     , 

vr 

100  years 

=  1  Century   .     .     , 

,    cen. 

yr.  mo.   da.    hr.     min.      sec. 
1  =  12  =  365  =  8760  =  525600  =  31536000 

Soak— 60,  60,  24,  365,  100. 

1.  In  most  business  computations  30  days  are  considered  a  month, 
and  12  months  a  year.    For  many  purposes  4  weeks  constitute  a  month. 

2.  The  common  year  contains  52  weeks  and  1  day,  the  leap  year  52 
weeks  and  2  days.     Hence,  commonly,  each  year  begins  one  day  later 
in  the  week,  but  the  year  succeeding  leap  year  begins  tico  days  later. 

3.  The  time  required  for  the  earth  to  revolve  around  the  sun  is  one 
year,  which  is  365  da.  5  hr.  48  min.  49.7  sec.,  or  very  nearly  365J  days. 
Instead  of  reckoning  this  part  of  a  day  each  year,  it  is  disregarded,  and 
an  addition  made  when  this  would  amount  to  one  day,  which  would 
be  very  nearly  every  fourth  year.     This  addition  of  one  day  is  made 
to  the  month  of  February.    Since  the  part  of  a  day  that  is  disregarded 
when  365  days  are  considered  as  a  year,  is  a  little  less  than  one-quarter 
of  a  day,  the  addition  of  one  day  every  fourth  year  is  a  little  too  much, 
and,  to  correct  this  excess,  addition  is  made  to  only  every  fourth  cen- 
tennial year.     With  this  correction  the  error  does  not  amount  to  much 
more  than  a  day  in  4000  years.     Therefore, 

Centennial  years  whose  number  is  exactly  divisible  by  400, 
and  other  years  whose  number  is  exactly  divisible  by  4,  are 
Leap  Tears. 


MEASUKES   OF   TIME.  183 

The  year  begins  with  the  month  of  January,  and  ends  with 
the  month  of  December. 

The  months,  their  names  and  the  number  of  days  in  each, 
are  as  follows : 


January,  31  da.     .     .  Jan. 

February,  28  or  29  da.  Feb. 

March,  31  da.     .     .  Mar. 

April,  30  da.     .     .  Apr. 

May,  31  da.     .     .  May. 

June,  30  da.    .    .  June. 


July,  31  da.  .  .  July. 

August,        31  da.  .  .  Aug. 

September,  30  da.  .  .  Sept. 

October,       31  da.  .  .  Oct. 

November,  30  da.  .  .  Nov. 

December,    31  da.  .  .  Dec. 


EXERCISES. 

1.  How  many  seconds  are  there  in  5  minutes?    In  6  min.  ? 

2.  How  many  minutes  are  there  in  ^  hour?     In  ^  hr.  ? 

3.  How  many  days  are  there  in  4  weeks?    In  5  wk.  ?    In 
8  wk.  ?     In  10  wk.  ? 

4.  How  many  hours  are  there  in  \  day  ?     In  ^  da.  ? 

5.  How  many  days  are  there  in  ^  year?     In  ^  month? 

6.  What  part  of  an  hour  are  30  minutes?     15  min.? 

7.  How  many  hours  are  there  in  90  minutes?     In  120 
min.  ?     In  240  min.  ? 

8.  How  many  seconds  are  there  in  5  hr.  15  min.  12  sec.  ? 

9.  How  many  seconds  are  there  in  6  hr.  27  min.  38  sec.  ? 

10.  Express  in  units  of  higher  orders  48695  sec. 

11.  Express  in  units  of  higher  orders  38497  sec. 

12.  How  many  minutes  are  there  in  5  yr.  of  365  da.  each  ? 

13.  How  many  days  are  there  from  Jan.  1st  to  May  1st? 

14.  How  many  days  are  there  from  April  1st  to  Oct.  15th  ? 

15.  Reduce  2  wk.  5  da.  13  hr.  to  hours. 

16.  Reduce  5  da.  10  hr.  15  min.  to  minutes. 

17.  Reduce  384600  sec.  to  higher  denominations. 

18.  Reduce  15  hr.  12  min.  18  sec.  to  seconds. 

19.  Reduce  32965  min.  to  higher  denominations. 


184 


DENOMINATE   NUMBEKS. 


CIRCULAR  OR  ANGULAR  MEASURE. 

275.  A  Circle  is  a  plane  surface,  bounded  by  a  curved 

line  every  point  of  which  is 
equally  distant  from  a  point 
within  called  the  Center. 

276.  The    Circumfer- 
ence is  the  line  that  bounds 

B   the  circle. 

277.  An  Arc  of  a  circle  is 
any  part  of  the  circumference. 

278.  A  Degree  is  ^  of 
the  circumference  of  a  circle. 

279.  The  Measure  of  an  Angle  is  that  part  of  the 
circumference  which  is  included  between  the  lines  which  form 
the  angle. 

Each  of  the  arcs  of  the  circumferences  ab,  cd,  DE,  is  a  measure 
of  the  same  angle,  and  therefore  contains  the  same  number  of  degrees ; 
but  since  each  degree  is  -3^  of  the  circumference,  the  length  of  a  de- 
gree must  vary. 

280.  Circular  or  Angular  Measure  is  used  to 
measure  arcs  of  circles  and  angles,  in  determining  latitude, 
longitude,  direction,  the  position  of  vessels  at  sea,  etc. 

TABLE. 

60  Seconds  (")  =  1  Minute     .     .     .     x 
60  Minutes          —  1  Degree     .     .     .     ° 
360  Degrees          =  1  Circumference  .     Cir. 

Cir.       °  " 

1  =  360  =  21600  =  1296000 


Scale— 60,  60,  360. 


MISCELLANEOUS.  185 

1.  A  Quadrant  is  J  of  a  circumference,  or  90°;    a  Semtant 
^  of  a  circumference,  or  60°. 

2.  The  length  of  a  degree  of  longitude  on  the  earth's  surface  at 
Equator  is  69.16  miles. 

3.  In  astronomical  calculations  30°  are  called  a  Sign,  and  there 
therefore  12  signs  in  a  circle. 


EXERCISES. 

1.  How  many  minutes  are  there  in  5°?     6°? 

2.  In  35  degrees  how  many  seconds  are  there?     In  27°? 
In  21°  12'  18"? 

3.  How  many  seconds  are  there  in  34°  12'  43"  ? 

4.  In  468560  seconds  how  many  minutes,  etc.,  are  there? 

5.  In  384500  seconds  how  many  minutes,  etc.,  are  there? 

6.  How  many  seconds  are  there  in  ^  Cir.  ?     In  ^?     In  ^? 

7.  How  many  minutes  are  there  in  2  quadrants?     In  2 
extants? 

COUNTING. 

281.  The  following  denominations  are   used  in   counting 
some  classes  of  articles : 

12  Things  =  1  Dozen    .     .     .     doz. 
12  Dozen   =  1  Gross      .     .     .    gr. 
12  Gross    =  1  Great  Gross     .     G.  gr. 

Two  things  are  often  called  a  pair,  six  things  a  set,  and  twenty  things 
a  score;  as  a  pair  of  birds,  a  set  of  spoons,  a  score  of  years. 


STATIONERS'  TABLE. 

282.  The  denominations  used  in  the  paper  trade  are : 

24  Sheets      =  1  Quire. 
20  Quires     =  1  Ream. 

2  Reams     —  1  Bundle. 

5  Bundles  =  1  Bale. 


186  DENOMINATE   NUMBERS. 

The  terms  folio,  quarto,  octavo,  applied  to  books,  indicate  the  number 
of  leaves  into  which  a  sheet  of  paper  is  folded.  Thus,  when  a  sheet 
of  paper  is  folded  into  2,  4,  8,  12,  16,  18,  or  24  leaves,  the  forms  are 
called  respectively,  folio,  4to,  or  quarto,  8vo,  or  octavo,  12nio,  16mo, 
18mo,  and  24mo. 

EXERCISES. 

1.  How  many  eggs  are  there  in  5  dozen?   7  doz. ?   10  doz.  ? 

2.  How  many  crayons  are  there  in  2  gross  ?     3  gr.  5  doz.  ? 

3.  How  many  things  are  there  in  a  great  gross  ? 

4.  What  will  be  the  cost  of  3  dozen  brushes  at  $.45  each  ? 

5.  A  man  lived  3  score  and  10  years.     What  was  his  age  ? 

6.  What  will  3  reams  of  paper  sell  for  at  $.15  per  quire? 


REDUCTION  OP  DENOMINATE  FRACTIONS. 

283.  The  principles,  processes  and  analyses  are  essentially 
the  same  as  those  of  denominate  integers. 

CASE    I. 

284.  To  reduce  denominate  fractions  to  equivalent 
numbers  of  lower  denominations. 

EXERCISES. 

1.  How  many  hours  are  there  in  1  day?     In  \  day?     In 
|  of  a  day?     In  f  of  a  day?     In  £  of  a  day? 

2.  How  many  ounces  are  there  in  -J-  pound  avoirdupois? 
In^lb.?     In|lb.? 

3.  How  many  pints  are  there  in  \  of  a  peck  ?     In  f  pk.  ? 

4.  How  many  pecks  and  quarts  are  there  in  f  of  a  bushel  ? 

5.  How  many  pounds  and  ounces  are  there  in  f  cwt.  ? 

6.  How  many  inches  are  there  in  f  of  a  foot  ?    f  ft.  ?   f  ft.  ? 


REDUCTION  OF  DENOMINATE  FKACTIONS. 


187 


7.  Change  -f-  of  a  rd.  to  units  of  lower  denominations. 

PROCESS.  ANALYSIS. — Since  in  1 

5   Of  ILL  yd.  =  |f  yd.  =  31f  yd.       yod  thf e  are  5i  ?ards> 

in  f  of  a  rod  there  will 

|f  Of    3       ft.    =   :    ||    ft.      :  =  2^  ft.  be    5   of    51     yards,  or    3|| 

|i  of  12  in.  =  ±f£  in.  =  9T6¥  in.        yards. 

Since  in  1  yard  there 
are  3  feet,  in  ££  of  a  yard  there  will  be  £J  of  3  ft,,  or  2JJ  ft. 

Since  in  1  foot  there  are  12  inches,  in  -J-J  of  a  foot  there  will  be  -JJ 
of  12  inches,  or  9T6¥  in. 

Therefore  f  of  a  rod  is  equal  to  3  yd.  2  ft.  9T6^  in. 

Change  the  following  to  lower  denominations: 


8.  -|  of  a  pound  Troy. 

9.  f  of  a  ton. 

10.  |  of  a  furlong. 

11.  of  an  acre. 


12.  |  of  a  peck. 

13.  T%  of  a  day. 

14.  -|  of  a  sq.  rd. 

15.  ^  of  a  cu.  yd. 


16.  Express  y^  of  a  gallon  as  a  fraction  of  a  gill. 


ANALYSIS. — Since  in  1  gallon  there  are  32  gills,  in  T J-j  of  a  gallon 
there  are  T^  of  32  gi.,  or  •££$  gi.    Hence  T^  gal.  =  -gfo  of  a  gill. 

17.  Express 

18.  Express 

19.  Express 

20.  Express 

21.  Express 

PROCESS. 
.685 
12 


8.220  oz. 

20 
4.400  pwt. 

24 
9.600  gr. 


O  of  a  bushel  as  a  fraction  of  a  pint. 
T  °f  a  m^e  as  a  fraction  of  a  foot, 
of  a  pound  as  a  fraction  of  a  scruple. 
.006  of  a  bushel  as  a  decimal  of  a  pint, 
in  lower  denominations  .685  of  a  pound  Troy. 

ANALYSIS. — Since  in  1  pound  there  are  12 
ounces,  in  .685  of  a  pound  there  are  .685  of  12 
ounces,  or  8.220  ounces. 

Since  there  are  20  pennyweights  in  1  ounce, 
in  .220  of  an  ounce  there  are  .220  of  20  penny- 
weights, or  4.400  pennyweights. 

Since  in  1  pennyweight  there  are  24  grains, 
in  .400  of  a  pennyweight  there  are  .400  of  24 
grains,  or  9.600  grains. 

Therefore  .685  Ib.  is  equal  to  8  oz.  4  pwt.  9.6  gr. 


188  DENOMINATE   NUMBERS. 

Express  in  units  of  lower  denominations: 


22.  £.575. 

23.  .  1935  of  a  pound  Troy. 

24.  .436    of  a  ream. 

25.  .1845  of  a  gallon. 


26.  .135    of  a  rod. 

27.  .455    of  a  mile. 

28.  .4832  of  a  bushel. 

29.  .684    of  a  league. 


CASE  IT. 

285.  To  change  denominate  fractions  to  equivalent 
fractions  of  higher  denominations. 


EXERCISES. 

1.  What  part  of  a  pound  Troy  is  1  ounce?     Is  •£  oz.?     Is 
ioz.? 

2.  What  part  of  a  ton  is  1  pound  ?     Is  £  Ib.  ?     Is  ^  Ib.  ? 

3.  What  part  of  a  mile  is  1  rod  ?     Is  |  rd.  ?     Is  £  rd.  ? 

4.  What  part  of  a  league  is  1  mile  ?     Is  1  rd.  ?     Is  -|-  rd.  ? 

5.  What  part  of  an  hour  is  •£  of  a  minute  ?   Is  |-  of  a  min.  ? 

6.  What  part  of  a  week  is  -f-  of  a  day  ?     -|  of  a  day  ? 

7.  What  part  of  a  bushel  is  -f  of  a  pint  ? 

PROCESS.  ANALYSIS.  —  Since  there  are  64  pints  in 

^   y  _i_  _  _s    hn  ^  bushel,  1  pint  is  ^  of  a  bushel,  and  f 
of  a  pint  is  f  of  &  of  a  bushel,  or  7f  ¥ 

Or,  of  a  bushel.     Or, 

g  Since  we  are  required  to  change  pints  to 

T2"  4  •  bushels  we  have  an  example  in  reduction 

T  2"  ~^~  *    """96"  P^-'  ascending,  and  hence  we  divide  by  2,  8,  and 

9T  -^  4   =  dhr  bu-  4,  respectively. 


8.  Reduce  -^  of  an  inch  to  the  fraction  of  a  yard. 

9.  Change  |-  of  a  second  to  the  fraction  of  an  hour. 

10.  Express  .375  of  a  week  as  a  fraction  of  a  year. 

11.  Express  .35  of  a  pound  as  a  fraction  of  a  ton. 

12.  Express  f  of  a  cubic  inch  as  a  fraction  of  a  cubic  foot. 


REDUCTION  OF  DENOMINATE  FRACTIONS.  189 

13.  Change  f  of  a  square  yard  to  a  fraction  of  an  acre. 

14.  Reduce  f  of  a  pint  to  a  fraction  of  a  barrel. 

CASE  III. 

286.   To  express  one  denominate  number  as  a  frac- 
tion of  another. 

1.  What  part  of  a  foot  are  3  in.  ?     6  in.  ?     9  in.  ? 

2.  What  part  of  an  hour  are  30  min.?     15  min.?     45 
min.? 

3.  What  part  of  a  gallon  is  1  pint?     2  pints?    1  quart? 

4.  What  part  of  a  gallon  are  2  quarts?     2  qt.   1  pt.? 
3  qt.  1  pt.  ? 

5.  What  part  of  3  ft.  6  in.  are  2  ft.  3  in.  ? 

ANALYSIS— Since  3  ft.  6  in.  =  42  in.,  and  2  ft.  3  in.  =  27  in.,  27  in. 

=  f|  of  42  in. 

6.  What  part  of  3  yd.  2  ft.  are  2  yd.  2  ft.  ? 

7.  What  part  of  5  gal.  3  qt.  1  pt.  are  2  gal.  1  qt.  1  pt.  ? 

8.  What  part  of  2  pounds  Troy  are  3  oz.  10  pwt.  ? 

9.  What  part  of  3  pecks  are  2  qt.  1  pt.  ? 

10.  What  part  of  3  barrels  are  13  gal.  3  qt.  2  pt.  2  gi.  ? 

11.  Express  15s.  7d.  in  the  decimal  of  a  pound  sterling: 

IST.  PROCESS.  ANALYSIS. — In  order  to  find  what  part 

.j  -     „-,  _         -\  0.7  (]         one  number  is  of  another,  both  must  be 

reduced  to  the  same  denomination.     15s. 

£1  =      240d.        7d,  =  187d.    and    <£l=:240d.     Therefore 

£-I|-Z-      =  £.7791  +     187d.  =  £Jf£,  which,  reduced  to  a  deci- 
mal, is  equal  to  £.7791  + .     Or, 

2c.  PROCESS.  ANALYSIS.  —  Since  7d.  is  T^  of  a  shil- 

i  9^7^  ling,  it  may  be  reduced  to  a  decimal  by 

annexing  ciphers  to  the  numerator  and 

.    .5833  +  s.  dividing   by   12,  which  gives   .5833  +  s. 

9  0  M  *  \  ft  Q  Q    i  Therefore  15s.  7d.  =  15.5833  +  s. 

Since    1    shilling    is   &   of    a   Pound> 
£.7791+  15.5833  +  s.  =  £  JL4,j$JLa  Or  £  .7791  + . 


190  DENOMINATE    NUMBERS. 

12.  Reduce  4  hr.  15  min.  to  the  decimal  of  a  day. 

13.  Reduce  3  pk.  2  qt.  to  the  decimal  of  a  bushel. 

14.  Reduce  3  ft.  6  in.  to  the  decimal  of  a  rod. 

15.  Reduce  18s.  5f  d.  to  the  fraction  of  a  pound. 

16.  Reduce  18s.  5fd.  to  the  decimal  of  a  pound. 

17.  Reduce  16  Ib.  11  oz.  to  the  fraction  of  a  hundred-weight. 

18.  Reduce  37  rd.  14  ft.  3  in.  to  the  decimal  of  a  mile. 

19.  Reduce  3  da.  5  hr.  14  min.  to  the  decimal  of  a  week. 

20.  Reduce  8  quires,  15  sheets,  to  the  decimal  of  a  ream. 

21.  Change  3  cd.  ft.  7  cu.  ft.  to  the  decimal  of  a  cord. 

22.  Change  654  yd.  9  in.  to  the  decimal  of  a  mile. 

23.  Change  4  oz.  7  pwt.  13  gr.  to  the  fraction  of  a  pound 
Troy. 

24.  Write  rules  for  each  of  the  cases  in  denominate  num- 
bers. 

REVIEW  EXERCISES. 

287.    1.  What  will  be  the  cost  of  15  Ib.  8  oz.  of  butter 
at  131  per  pound? 

2.  What  must  be  paid  for  3  pk.  2  qt.  of  berries  at  9  cents 
a  quart? 

3.  Mr.  A.  sold  18  bu.  3  pk.  of  barley  at  $1.05  per  bushel. 
How  much  did  he  get  for  it? 

4.  How  much  must  be  paid  for  making  42  rd.  7  ft.  8  in. 
of  fence  at  $  .75  per  foot? 

5.  How  much  butter  at  $  .30  a  pound  must  be  given  for 
12  gal.  3  qt.  of  molasses,  at  $  .50  per  gallon? 

6.  Bought  15  bu.  of  oats  at  $  .37^  a  bushel,  and  sold  them 
at  15  cents  a  half-peck.     How  much  did  I  gain? 

7.  How  many  cords  of  wood  are  there  in  a  pile  4  ft.  wide, 
6  ft,  high,  60  ft.  long?     What  would  it  cost  at  84.25  a  cord? 

8.  A  man  built  a  cistern  10  ft.  long  and  6  ft.  wide,  that 
would  hold  100  barrels.     How  high  did  he  make  it? 


REVIEW   EXERCISES.  191 

9.  What  is  a  druggist's  profit  if  he  buys  opium  at  $.75  per 
ance  Avoirdupois,  and  sells  it  at  $1  per  ounce  Troy? 

10.  What  are  the  contents  of  a  field  15  rd.  8  ft.  wide,  27 
9  ft.  long?     What  is  its  value  at  $150  per  acre? 

11.  How  many  days  of  10  hours  each  will  it  require  to 
ake  a  million  marks  if  I  make  2  per  second? 

12.  What  is  the  value  of  a  plank  18  ft.  long,  16  in.  wide, 
ad  4  in.  thick,  at  $18  per  M? 

13.  If  at  10  cents  a  foot  the  Atlantic  cable  cost  $1689600, 
vhat  is  its  length? 

14.  A  druggist  put  up  7§  83  49  in  two-grain  pills.     How 
any  pills  did  he  put  up? 

15.  Bought  paper  at  $2.55  per  ream  and  sold  it  at  20 
ents  per  quire.     How  much  did  I  gain? 

16.  How  much  sugar  at  12  cents  a  pound  can  be  obtained 
br  13  Ib.  7  oz.  butter  at  27^-  cents  a  pound? 

17.  A  farmer  sold  3  piles  of  wood  at  $4.60  per  cord.     The 
following  are  the  dimensions  of  the  piles:  The  first  was  73 
ft.  9  in.  long,  6  ft.  high,  and  4  ft.  wide;  the  second  was  30 
ft.  long,  7  ft.  2  in.  high,  and  4  ft.  wide;  the  third  was  37  ft. 
long,  3  ft.  6  in.  high,  and  4  ft.  wide.     How  much  should  he 
receive  for  his  wood? 

18.  A  printer  used  4  reams  8  quires  12  sheets  of  paper  for 
half-sheet  posters.      How  many  did  he  print?      What  did 
they  cost  at  $6.50  per  M? 

19.  Hay  at  $18  per  ton  is  exchanged  for  flour  at  $6.85  per 
barrel.     How  many  barrels  are  equal  to  a  ton? 

20.  Two  men  who  are  equal  partners,  obtained  from  a  field 
327  bu.  3  pk.  5  qt.  of  oats.     One  of  them  claimed  167  bu. 
3  pk.  for  his  share.     Did  he  claim  too  much  or  too  little? 
How  much? 

21.  A  cubic  foot  of  water  weighs  about  62  Ib.  8  oz.     What 
will  be  the  weight  or  pressure  on  a  square  yard  where  the 
sea  is  20  fathoms  deep? 


192  DENOMINATE   NUMBEES. 


ADDITION. 

288.  The  processes  of  adding,  subtracting,  multiplying, 
and  dividing  compound  numbers  are  based  upon  the  same 
principles  as  those  governing  similar  operations  in  simple 
numbers. 

The  only  difference  between  the  processes  is  caused  by  com- 
pound numbers  having  a  varying  scale,  while  simple  numbers 
have  a  uniform  decimal  scale. 

EXERCISES. 

1.  What  is  the  sum  of  130  rd.  5  yd.  1  ft.  6  in.,  215  rd. 
2  ft.  8  in.,  304  rd.  4  yd.  11  in.? 

PROCESS.  ANALYSIS. — The  numbers  should 

-,  -,     ,  f.  be  written  as  in  simple  addition,  so 

ra.        yd.      it.       in. 
.  OA        IT         -I         n       that  units  of  the  same  denomma- 

tion  stand  in  the  same  column,  and 
for  convenience  we  begin  at  the 
304  4  0  11  right  to  add. 


2  mi.       10        4|-     2        1  The  sum  of  the  inches  is  25  in., 


/)  i  _  i         a        which  ig  equal  to  2  ft.  1  in.     We 

_>  _  "2"~  write  the  1  under  the  inches  and 

2  mi.       10        5        0        7        add  the  2  ft.  with  the  feet.     The 

sum  of  the  feet  is  5  ft.,  or  1  yd.  2  ft. 

We  write  the  2  as  feet  in  the  sum  and  add  the  1  yd.  with  the  yards. 
The  sum  of  the  yards  is  10  yd.,  or  1  rd.  4J  yd.    We  write  the  4J  yd. 

as  yards  of  the  sum,  and  add  the  1  rd.  with  the  rods.     The  sum  of  the 

rods  is  650  rd.,  or  2  mi.  10  rd.,  which  we  write  as  miles  and  rods  of  the 

sum. 

Therefore  the  sum  is  2  mi.  10  rd.  4£  yd.  2  ft.  1  in.      Or,  since  \  yd. 

equals  1  ft.  6  in.,  the  sum  may  be  expressed  as  2  mi.  10  rd.  5  yd.  7  in. 

RULE.  —  Change  the  rule  for  the  addition  of  simple  numbers 
so  that  it  may  be  applicable  to  denominate  numbers. 


ADDITION.  193 

2.  What  is  the  sum  of  12  Ib.  5  oz.  13  pwt.,  21  Ib.  8  oz. 
15  pwt.,  13  Ib.  7  oz.  10  pwt.,  51  Ib.  3  oz.  17  pwt? 

3.  What  is  the  sum  of  £71  6s.  5|d.,  £32  8s.  5£d.,  £61 
15s.  Sid.,  £37  18s.  5fd.,  £115  lls.  7d.? 

4.  Find  the  sum  of  10  mi.  217  rd.  2  yd.  3  ft.  4  in.,  7  mi. 
185  rd.  3  yd.  9  in.,  19  mi.  37  rd.  6  yd. 

5.  Find  the  sum  of  3  T.  7  cwt.  39  Ib.  8  oz.,  8  T.  11  cwt. 
48  Ib.,  11  oz.,  13  T.  33  Ib.  10  oz.,  9  cwt.  18  Ib.  9  oz. 

6.  Find  the  sum  of  18  gal.  3  qt.  1  pt.  3  gi.,  15  gal.  2  qt. 

1  pt.  2  gi.,  11  gal.  2  qt.  2  gi.,  3  qt.  1  pt.  1  gi. 

7.  A  miller  bought  four  loads  of  grain  containing,  respect- 
ively, 25  bu.  3  pk.,  28  bu.  2  pk.,  32  bu.  3  pk.  5  qt.,  28  bu. 

2  pk.  7  qt.     How  much  grain  did  he  buy  ? 

8.  How  much  wood  is  there  in  3  piles  containing,  respect- 
ively, 37  C.  21  cu.  ft.  1140  cu.  in.,  29  C.  110  cu.  ft.  708 
cu.  in.,  and  34  C.  121  cu.  ft.  398  cu.  in.? 

9.  Find  the  sum  of  f  mi.,  .35  rd.  and  2f  rd. 

PROCESS. 

rd.  ft.  in. 

o      .          -,OP7  o  A  9              ANALYSIS.  — Each    of    the 

f  mi.  —  137  2  44  f     ^        .                -,  .     . 

7  7  fractions   is  expressed  in  in- 

.00  rd.  -  "TO        tegers  of  lower  denominations, 

2 1  rd.  =        2        6       2^          and  then  they  are  added. 

139    14 

10.  A  merchant  sold  12f  yards  of  cloth  to  one  person,  8f 
yards  to  another,  37^-  yards  to  another,  39f  yards  to  another. 
How  many  yards,  feet  and  inches  did  he  sell? 

11.  What  is  the  amount  of  land  in  the  following  lots,  the 
first  containing  -J-  of  an  acre,  the  second  f  of  an  acre,  the 
third  1291  sq.  rd.,  and  the  fourth  118£  sq.  rd.  ? 

12.  A  merchant  sold  the  following  quantities  of  molasses, 
viz:    On  June  15,  24  gal.  2  qt.  3  pt. ;    June  16,  45^  gal.; 
June  17,  li  bbl.  (39f  gal.)     How  much  did  he  sell  in  that 
time? 

13 


194  DENOMINATE   NUMBEKS. 

13.  James  is  3  yr.  4  mo.  18  da.  old,  Henry  is  2  yr.  8  mo. 
6  da.  older  than  James,  William  is  7  yr.  10  mo.  24  da.  older 
than  Henry,  and  Herbert  is  20  mo.  older  than  William.    How 
old  is  Herbert? 

14.  Find  the  sum  of  20£  cwt.,  16|  T.,  17£  lb.,  19  cwt. 
18  lb.  7  oz.,  15  lb.  8  oz.,  2  T.  7  lb.  5  oz.,  f  lb.,  f  T.,  2  T. 
3  cwt.  57  lb.  4  oz. 


SUBTRACTION. 

289.    1.  Prom  127  rd.  3  yd.  1  ft.  7  in.,  subtract  100  rd. 
4  yd.  2  ft.  9  in. 

PROCESS.  ANALYSIS.  —  The  numbers  should  be 

rd.        yd.     ft.      in         written    as   in   simple   subtraction,   so 

.  ~  7        o        i        7         that  units  of  the  same  order  stand  in 

the  same  column,  and,  for  convenience, 

begin  at  the  right  to  subtract. 

26       31     1     1  0  Since  we  can  not  subtract  9  in.  from 

Q  l_-j        a         7  in.,  we  unite  with  7  in.  a  unit  of  the 

'  next  higher  order,  making  1  ft.  7  in., 


26  4  0  4  or  19  in.  Then  9  in.  from  19  in.  leaves 
10  in.,  which  we  write  as  inches  in  the 

remainder.  Inasmuch  as  1  ft.  was  united  with  7  in.,  there  are  no  feet 
remaining  in  the  minuend. 

Since  we  can  not  subtract  2  ft.  from  0  ft.,  we  unite  with  0  ft,  a  unit 
of  the  next  higher  order,  making  3  ft.  Then  2  ft.  from  3  ft.  leaves  1  ft,, 
which  we  write  as  the  feet  of  the  remainder. 

Since  4  yd.  can  not  be  subtracted  from  2  yd.,  we  unite  with  2  yd.  a 
unit  of  the  next  higher  order  and  proceed  as  before.  The  remainder 
will  be  26  rd.  4  yd.  0  ft.  4  in. 

RULE.  —  Change  the  rule  for  subtraction  of  simple  numbers  so 
that  it  may  be  applicable  to  denominate  numbers. 

2.  From  2  mi.  116  rd.  4  yd.  0  ft.  4  in.,  take  1  mi.  120  rd. 
2  yd.  1  ft.  8  in. 


SUBTRACTION.  195 

3.  From  15  cwt.  37  Ib.  10  oz.,  take  8  cwt.  42  Ib.  8  oz. 

4.  From  1  hhd.  38  gal.  3  qt.  2  pt.,  take  60  gal.  2  qt.  1  gi, 

5.  From  13  Ib.  8  oz.  13  pwt.  15  gr.,  take  8  Ib.  8  oz.  16  pwt. 
15  gr. 

6.  From  18°  33'  16",  take  9°  42'  28". 

7.  From  37  C.  7  cd.  ft.  11  cu.  ft.,  take  18  C.  7  cd.  ft. 
12  cu.  ft. 

8.  From  f  bbl.  take  7|  gal. 

PROCESS. 

gal.      qt.     pt.       gi.  ANALYSIS. — The   fractions 

3.  bbl.  =2321  are  nrst  expressed  in  integers 

7—  ffal    =7        2       0       3—       °^  l°wer  denominations  and 


then  subtracted. 


16      0      0 


9.  From  f  of  an  acre  of  land  a  piece  containing  72  sq.  rd. 
160  sq.  ft.  39  sq.  in.  was  sold.     How  much  was  left? 

10.  A  merchant  sold  cloth  for  £384  6s.  5|d.  which  cost 
him  £297  9s.  8|d.     How  much  was  his  profit? 

11.  From  a  farm  of  285  acres  there  were  sold  at  one  time 
97f  acres,  and  at  another  38  A.  39 \  sq.  rd.     How' much  was 
left? 

12.  A  merchant  bought  9  reams  18  quires  15  sheets  of 
paper,  from  which  he  sold  3^  reams.     How  much  remained 
unsold? 

13.  How  long  was  it  from  Jan.  10,  1841,  to  May  7,  1853? 

PROCESS.  ANALYSIS. — Since  the  later  date  expresses 

1853        5  7        *ne  £reater  PeriO(l  °f  time,  we  write  it  as  the 

1  &  J.  1        1        10       minuend,  and  the  earlier  date  as  the  subtra- 

hend,  giving  the  month  its  number  instead 

12        3        27        of  the  name.     We  then  subtract  as  in  de- 
nominate numbers,  considering  30  days  one 

month,  and  12  months  one  year.     The  remainder  will  be  the  time  as 
correct  as  it  can  be  expressed  in  months  and  days. 


14.  How  long  was  it  from  Jan.  3,  1843,  to  Mar.  15,  1851? 


196  DENOMINATE   NUMBERS. 

15.  How  old  was  a  man  who  was  born  April  2,  1803,  and 
who  died  Dec.  15,  1869? 

16.  A  man  bought  a  farm  May  15,  1860,  and  paid  for  it 
Jan.  5,  1871.     How  long  did  it  take  to  pay  for  it? 

17.  A  legacy  of  $3000  was  to  be  paid  to  a  man  3  yr.  2  mo. 
5  da.  after  Dec.  8,  1837.     When  was  it  to  be  paid? 

18.  How  many  years,  months  and  days  from  the  day  of 
your  birth?  or,  How  old  are  you? 

19.  The  American  Civil  War  began  April  11,  1861,  and 
ended  April  9,  1865.     How  long  did  it  continue? 

20.  A  note  dated  July  9,  1871,  was  paid  October  10,  1876. 
How  long  did  it  run  before  it  was  paid? 


MULTIPLICATION. 

290.    1.  How  much  is  5  times  147  rd.  4  yd.  2  ft.  8  in.? 

PROCESS.  ANALYSIS. — We  write  the  numbers  as 

rd,     yd.  ft.   in.       *n  simple  numbers,  and  for  convenience 
147     4     2     8       begin  at  the  right  to  multiply. 

P-  5  times  8  in.  are  40  in.,  or  3  ft.  4  in. 

We  write  the  4  in.  as  inches  in  the  prod- 

2  mi.     9  9     2     1      4       uct,  and  reserve  the  3  ft.  to  add  with  the 

product  of  feet. 

5  times  2  ft.  are  10  ft;  10  ft.  +  3  ft.  reserved  equal  13  ft.,  or  4  yd. 
1  ft.  We  write  the  1  ft.  in  the  product  and  reserve  the  4  yd.  to  add 
to  the  product  of  yards. 

5  times  4  yd.  equal  20  yd. ;  20  yd.  +  4  yd.  reserved  equal  24  yd.,  or 
4  rd.  2  yd.  We  write  the  2  yd.  in  the  product  and  reserve  the  rods  to 
add  to  the  product  of  rods. 

5  times  147  rd.  are  735  rd.;  735  rd.  +  4  rd.  reserved  equal  739  rd., 
or  2  mi.  99  rd.,  which  we  write  in  the  product. 

Therefore  the  product  is  2  mi.  99  rd.  2  yd.  1  ft.  4  in. 

RULE. — Modify  the  rule  for  multiplication  of  simple  numbers 
so  that  it  may  be  applicable  to  denominate  numbers. 


DIVISION.  197 

2.  Multiply  9  gal.  3  qt.  1  pt.  3  gi.  by  7. 

3.  Multiply  17  Ib.  8  oz.  3  pwt.  15  gr.  by  9.  , 

4.  Multiply  1  T.  4  cwt.  35  Ib.  6  oz.  by  10. 

5.  A  farm  consists  of  7  fields  each  containing  18  A.  25 
sq.  rd.     How  much  land  does  it  comprise? 

6.  What  is  the  length  of  a  fence  which  encloses  a  square 
field  each  side  of  which  is  28  rd.  5  yd.  2£  ft.  long? 

7.  How  much  wood  is  there  in  7  piles,  each  containing 
130.  7cd.  ft.  24  cu.  ft.? 

8.  What  will  14^  yd.  of  lace  cost  at  £2  5s.  6d.  per  yard? 

9.  What  is  the  value  of  4  loads  of  potatoes,  each  contain- 
ing 27  bu.  3  pk.,  at  $.45  per  bushel? 


DIVISION. 

291.    1.  Divide  27  bu.  3  pk.  5  qt.  1  pt.  into  6  equal  parts. 
PROCESS.  ANALYSIS. —  Since  the  quan- 

6  )  2  7  bu.  3  pk.  5  qt.  1   pt.      ^ is  to  be  divided  »*> 6  e1uf l 

parts,   each   part   will    contain 

^  1~6"  one-sixth  of  the  quantity. 

One-sixth  of  27  bu.  is  4  bu. 

with  a  remainder  of  3  bu.  We  write  the  4  bu.  in  the  quotient  and 
unite  the  3  bu.  remaining  with  the  number  of  the  next  lowest  denomi- 
nation, making  15  pk. 

One-sixth  of  15  pk.  is  2  pk.  and  3  pk.  remaining.  We  write  the 
2  pk.  in  the  quotient,  and  unite  the  3  pk.  remaining  with  the  number 
of  the  next  lower  denomination,  making  29  qt. 

One-sixth  of  29  qt.  is  4  qt.  and  5  qt.  remaining.  We  write  the 
4  qt.  in  the  quotient,  and  unite  the  5  qt.  with  the  number  of  the  next 
lowest  denomination,  making  11  pt. 

One-sixth  of  11  pt.  is  If  pt.,  which  we  write  in  the  quotient. 

Therefore  the  quotient  is  4  bu.  2  pk.  4  qt.  If  pt. 

RULE. — Change  the  rule  for  the  division  of  simple  numbers  so 
that  it  may  be  applicable  to  denominate  numbers. 


198  DENOMINATE    NUMBERS. 

2.  In  8  bags  there  are  17  bu.  3  pk.  4  qt.     How  much  does 
each  bag  contain  ? 

3.  A  gentleman  divided  his  farm  of  427  A.  131  sq.  rd. 
equally  among  his  5  sons.     What  was  the  share  of  each  ? 

4.  A  brewer  filled  4  casks  of  equal  size  from  a  vat  con- 
taining 315  gal.  3  qt.     How  large  was  each  cask? 

5.  16  T.  1300  Ib.  of  hay  was  drawn  at  9  loads.     What 
was  the  average  weight  per  load? 

6.  If  a  pile  of  wood  containing   8  cords  100  cu.  ft.   be 
equally  divided  among  3  persons,  how  much  will  each  receive  ? 

7.  When  £31  5s.  8d.  is  divided  equally  among  10  persons, 
how  much  does  each  receive? 

8.  If  31  cwt.  18  Ib.  of  tea  is  put  up  in  packages,  each 
containing  3  Ib.  8  oz. ,  how  many  packages  will  there  be  ? 

PROCESS.  ANALYSIS.— Since 

31  cwt.  18  Ib.  =49888  oz.      the.  divisor   and  .the 

dividend    are   similar 

3  Ib.  8  oz.  =  56    oz.        denominate    numbers, 

4  9  8  8  8  oz.  •+-  5  6   oz.  =        8  9  Of  we  may  reduce  them 

to  the  same  denomi- 
nation, and  then  proceed  to  divide  as  in  simple  numbers. 

9.  How  many  times  must  a  man  dip  with  a  dipper  hold- 
ing 2  qt.  1   pt.    so  that  he   may  empty  a  cask  containing 
31  gal.  ? 

10.  If  a  man  walks  at  an  average  rate  of  23  mi.  160  rd.  4  yd. 
2  ft.  per  day,  how  long  will  it  take  him  to  walk  100  miles? 

11.  If  a  man  can  travel  300  mi.  in  13  days,  how  far  can 
he  travel  daily? 

12.  How  many  barrels  of  sugar,  each  containing  2  cwt. 
35  Ib.,  are  there  in  3  T.  4  cwt.  18  Ib.? 

13.  How  many  spoons,  each  weighing  2  oz.  10  pwt.,  can 
be  made  from  13  Ib.  7  oz.  15  pwt.  of  silver? 

14.  How  many  pickets  2  ft.  4  in.  long  and  2  in.  wide,  can 
be  made  out  of  5  boards  each  11  ft.  8  in.  long  arid  8  in.  wide? 


LONGITUDE   AND    TIME.  199 


LONGITUDE  AND  TIME. 

292.  1.  Where  does  the  sun  appear  to  rise? 

2.  How  long  will  it  be  before  it  rises  again? 

3.  Through  how  many  degrees  of  space  does  it  appear  to 
pass  in  this  daily  motion?  Ans.  360°. 

4.  Since  it  seems  to  travel  360°  in  one  day,  or  24  hours, 
how  great  will  be  its  apparent  motion  in  1  hour? 

5.  If  the  earth  moves  15°  in  1  hour,  how  far  will  it  move 
in  1  minute? 

6.  If  it  moves  15'  in  one  minute  of  time,  how  far  will  it 
move  in  1  second? 

7.  How  does  the  number  of  degrees  passed  over  compare 
with  the  number  of  hours?     The  number  of  minutes  of  space 
with  the  number  of  minutes  of  time  ?     The  number  of  seconds 
of  space  with  the  number  of  seconds  of  time? 

8.  When  it  is  sunrise  at  New  York,  how  long  will  it  be 
before  it  is  sunrise  at  a  place  15°  west  of  New  York?     30° 
west?     45°  west?     60°  west? 

9.  When  it  is  sunrise  at  New  York,  how  long  before  was 
it  sunrise  at  a  place  15°  east?     30°  east?     45°  east? 

10.  When  it  is  sunrise  at  any  place,  how  long  will  it  be 
before  it  is  sunrise  at  a  place  15°  west?    15°  east?    30°  west? 
30°  east? 

11.  When  it  is  noon  at  any  place,  what  time  is  it  at  a  place 
15°  west?     15°  east?     30°  west?     30°  east? 

12.  If  I  travel  eastward  will  my  watch  become  too  slow  or 
too  fast?     If  I  travel  westward  what  change  will  take  place? 

13.  What  places  have  sunrise  at  the  same  time?     Noon 
at  the  same  time?     Midnight  at  the  same  time? 

293.  A  Meridian,  is  an  imaginary  line  passing  from  the 
North  Pole  to  the  South  Pole  through  any  place. 


200  DENOMINATE   NUMBERS. 

294:.   Longitude  is  the  distance  east  or  west,  from  a 
given  meridian. 

KELATION  BETWEEN  LONGITUDE  AND  TIME. 

15°  of  Longitude  make  1  Hour  difference  in  time. 
15 /  of  "  make  1  Minute  difference  in  time. 
15 /x  of  "  make  1  Second  difference  in  time. 

1°    of          "           makes  4  Minutes  difference  in  time. 

1'  of         "          makes  4  Seconds  difference  in  time. 


WRITTEN    EXERCISES. 

1.  The  longitude  of  Boston  is  71°  3'  30"  west;  that  of  Ch> 
cinnati,  84°  29'  31"  west.     What  is  the  difference  in  time  ? 

PROCESS.  ANALYSIS. — We  first  find  the  differ- 

o     o  Q'     Q  1  "       ence  *n  l°ngitude  °^  ^e  two  places>  an(l 
since  there  are  15  times   as  many  de- 

7  1  grees,  minutes  and  seconds  as  there  are 

1  5  ^  1  3         26          1          hours,  minutes  and  seconds  of  time,  we 

~T~3       T-T-       find  ^  of  13°  26'  1",  which  is  53  min. 

44  sec.,  the  difference  in  time. 

2.  When  it  is  12  o'clock  M.  at  Philadelphia  it  is  5  o'clock, 
10  min.  P.  M.  at  Paris.     What  is  the  longitude   of   Paris, 
the  longitude  of  Philadelphia  being  75°  10'  west? 

PROCESS.  ANALYSTS. — Since  in   1 

5  hr.  1  0    min.  hour  the  earth  moves  15° 

15  of  distance,  in  1  minute  157 

-  -  o      ^77T7   T^  .     -r  of  distance,  in  1  second  15X/ 
77°     30'  difference  m  Long.         of  distanc<;  the  difference 

*  **  in    longitude    will    be    15 
2  °      20'  east,  Long,  of  Paris.        times  as  many  degrees,  min- 
utes and  seconds  of  distance 

as  there  are  hours,  minutes  and  seconds  of  time.  Since  Philadelphia 
is  75°  10X  west,  and  the  difference  is  77°  3CK,  the  longitude  of  Paris 
is  2°  207  east. 


LONGITUDE    AND    TIME.  201 

To  find  the  difference  in  time  when  the  difference  in  longi- 
tude is  given: 

Divide  the  difference  in  longitude,  expressed  in  degrees,  etc. ,  by 
15;  the  several  quotients  will  be  the  difference  in  time  in  hours, 
minutes,  and  seconds. 

To  find  the  difference  in  longitude  when  the  difference  in 
time  is  given: 

Multiply  the  difference  in  time,  expressed  in  hours,  minutes  and 
seconds,  by  15;  the  several  products  will  be  the  difference  in  longi- 
tude, in  degrees,  minutes,  and  seconds. 

3.  Two  places  are  32°  IS'  24"  apart.     What  is  the  differ- 
ence in  time  between  them? 

4.  When  it  is  noon  at  San  Francisco  it  is  3  hr.  9  min. 
7  sec.  P.  M.  at  Philadelphia.     What  is  the  longitude  of  San 
Francisco  if  that  of  Philadelphia  is  75°  10'  west? 

5.  New  York  is  74°  3'  west  longitude,  and  Paris,  France, 
is  2°  20'  east.     How  much  earlier  is  it  sunrise  in  Paris  than 
in  New  York? 

6.  Washington  is  77°  west  of  Greenwich,  England.    What 
is  their  difference  in  time? 

7.  When  it  is  noon  at  Washington,  which  is  77°  west, 
what  time  is  it  at  New  York,  which  is  74°  3'  west? 

8.  The  difference  in  time  between  Halifax,  Nova  Scotia, 
and  Charleston,  S.  C.,  is  1  hr.  5  min.  8  sec.     What  is  their 
difference  in  longitude? 

9.  Pekin,  China,  is  116°  27'  30"  east  longitude,  and  Wash- 
ington is  77°  west  longitude.     When  it  is  noon  on  January 

st  at  Washington,  what  time  is  it  at  Pekin? 

10.  A  gentleman  traveling  found,  on  arriving  at  his  des- 
nation,  that  his  watch,  which  kept  correct  time,  was  1  hr. 

min.  slow.     Which  way  was  he  traveling  ?     How  far  had 
traveled  ? 


295.  The  Metric  System  of  weights  and  measures 
has  been  legalized  by  the  United  States,  most  of  the  coun- 
tries of  Europe,  and  several  countries  of  Central  and  South 
America. 

Although  this  system  is  extremely  valuable  on  account  of  its  sim- 
plicity, it  is  not  in  general  use  in  this  country,  and  hence  is  not  treated 
as  fully  here  as  the  other  divisions  of  Denominate  Numbers. 

296.  The  Unit  of  Length,  called  the  Metre  (meeter), 
from  which  the  system  derives  its  name,  is  nearly  one  ten- 
millionth  of  a  quadrant  of  the  earth's  circumference. 

297.  The  Unit  of  Area,  called  the  Are  (air),  is  a 
square  whose  side  is  10  metres.     It  contains  100  square 
metres. 

298.  The  Unit  of  Solidity,  called  the  Stere  (stair), 
is  a  cube  whose  edge  is  one  metre. 

299.  The  Unit  of  Capacity,  called  the  Litre  (leeter), 
contains  a  volume  equal  to  that  of  a  cube  whose  edge  is  one- 
tenth  of  a  metre. 

300.  The  Unit  of  Weight,  called  the  Gramme,  is 

the  weight  of  a  cube  of  distilled  water  whose  edge  is  one- 
hundredth  of  a  metre. 

It  must  be  weighed  in  a  vacuum  and  at  the  period  of  its  greatest 
density,  39.2  Fahrenheit. 
(202) 


METRIC    SYSTEM.  203 

301.  From  these  standard  units  are  derived  the  multiples 
and  sub-multiples  which  are  named  to  express  units  of  higher 
or  lower  orders  in  the  decimal  scale.  Thus, 

For  multiples,  Greek  numerals  are  used: 

Deka,  10;    Hecto,  100;    Kilo,  1000;   Myria,  10000. 

For  sub-multiples  the  Latin  ordinals  are  used: 
Deci,  10th;   Centi,  100th;    Milli,  1000th. 

Dekametre  .    .    .     .     .  means  10  Metres. 

Dekagramme    ....        "  10  Grammes. 

Hectometre "  100  Metres. 

Kilolitre "  1000  Litres. 

Myriagramme  ....  .  10000  Grammes. 

Centigramme    ....        "  TJ^  Gramme. 

Milligramme    .     .    -     .        "  j^  Gramme. 


MEASURES  OF  EXTENSION. 

302.  The  Metre  is  the  unit  of  length. 

TABLE. 

10  Millimetres  =  1  Centimetre    =      .3937079  in. 

10  Centimetres  =  1  Decimetre     =     3.937079  in. 

10  Decimetres  =  1  Metre         —  39.37079  in. 

10  Metres  —  1  Dekametre    ==£  32.808992  ft. 

10  Dekametres  «  1  Hectometre  =c  19.927817  rd. 

10  Hectometres  =  1  Kilometre     =       .6213824  mi. 

10  Kilometres  =  1  Myriametre  =  6.213824  mi. 

303.  The  Are  is  the  unit  of  land  measure. 

TABLE. 

1  Centiare   =  1  Sq.  Metre      =       1.196034  sq.  yd. 
100  Centiares  =  1  Are  =  119.6034  sq.  yd. 

100  Ares          =.  1  Hectare         —       2.47114  acres. 


204  DENOMINATE    NUMBERS. 

304.  The   Square   Metre  is  the  unit  for  measuring 
ordinary  surfaces,  as  flooring,  ceiling,  etc. 

TABLE. 

100  Sq.  Millimetres  ==  1  Sq.  Centimetre  —       .155  +  sq.  in. 
100  Sq.  Centimetres  =  1  Sq.  Decimetre    =  15.5  +  sq.  in. 
100  Sq.  Decimetres    =  1  Sq.  Metre      =     1.196  +  sq.  yd. 

305.  The  Stere  is  the  unit  of  wood  and  solid  measure. 

TABLE. 

1  Decistere  =  3.531  +  cu.  ft. 
10  Decisteres  =  1  Stere  =  35.316  +  cu.  ft, 
10  Steres  =  1  Dekastere  =  13.079  +  cu.  yd. 

306.  The  Cubic  Metre  is  the  unit  for  measuring  many 
ordinary  solids;  as  excavations,  embankments,  etc. 

TABLE. 

1000  Cu.  Millimetres  =  1  Cu.  Centimetre  =      .061  +  cu.  in. 
1000  Cu.  Centimetres  =  1  Cu.  Decimetre    =  61.026  cu.  in. 
1000  Cu.  Decimetres  ==  1   Cu.  Metre.    =  35.316  cu.  ft. 


MEASURES  OF  CAPACITY. 

307.  The  Litre  is  the  unit  of  capacity,  both  of  liquid 
and  dry  measure.     It  contains  about  a  quart,  liquid  measure. 

TABLE. 

10  Millilitres  ===  1  Centilitre    =  .6102  cu.  in.  =  .338  fluid  oz. 

10  Centilitres  =£  1  Decilitre     =  6.1022  cu.  in.  =  .845  gills. 

10  Decilitres  *=  1  Litre        =  .908  quart  =  1.0567  qt. 

10  Litres  =  1  Dekalitre    =  9.08  quarts  ==  2.6417  gal. 

10  Dekalitres  ===  1  Hectolitre  =  2.8372  +  bu.  =  26.417  gal. 

10  Hectolitres  =  1  Kilolitre     =  28.372  +  bu.  ==  264.17  gal. 

10  Kilolitres  =*  1  Myrialitre  *?  283.72  +  bu.  =  2641.7  + gal. 


METRIC    SYSTEM. 


205 


MEASURES  OF  WEIGHT. 
308.  The  Gramme  is  the  unit  of  weight. 


TABLE. 


10  Milligrammes 
10  Centigrammes 
10  Decigrammes 
10  Grammes 
10  Dekagrammes 

=  1  Centigramme     = 
=  1  Decigramme      = 
=  1  Gramme      = 
=  1  Dekagramme     — 
=  1  Hectogramme    = 

.15432  +  gr. 
1.54324+  " 
15.43248+  " 
.35273  +  oz.  Av. 
3.52739+  "     " 

10  Hectogrammes 

=  1{KOT°jSfeT}  = 

2.20462  +  lb.    " 

10  Kilogrammes 
10  Myriagrammes, 
100  Kilogrammes 
10  Quintals,  or 
1  000  Kilogrammes 

=  1  Myriagramme   — 
1  ==  1  Quintal 

~|           f  Tonneau,  or  ") 
}=   l{       Ton       r 

22.04621+  "     " 
2204.62125+  "     " 

The  Kilogramme,  or  Kilo,  is  the  unit  of  common 
weight  in  trade,  and  is  a  little  less  than  2^  Ib.  Avoirdupois. 

The  Tonneau  is  used  for  weighing  very  heavy  articles, 
and  is  about  204^  Ib.  more  than  a  ton. 


EXERCISES. 

1.  What  metric  unit  corresponds  most  nearly  to  our  yard? 
How  many  metres  are  there  in  a  rod? 

2.  What  metric  measure  corresponds  most  nearly  to  our 
mile? 

3.  What  unit  expresses  nearly  one  ton? 

4.  What  unit  is  nearly  equal  to  one  quart? 

5.  Reduce  45  dekagrammes  to  grammes. 

6.  Express  7  dekametres,  25  centimetres,  as  metres. 

7.  How  many  litres  are  there  in  a  kilolitre?     In  a  deka- 
litre? 


206  DENOMINATE   NUMBERS. 

8.  What  is  the  value  of  an  acre  in  metric  units? 

9.  What  metric  measure  corresponds  most  nearly  to  one 
bushel? 

10.  Reduce  586.431  metres  to  decimetres. 

11.  What  will  be  the  cost  of  324.16  hectogrammes  of  sugar 
at  22  cents  per  hectogramme? 

12.  A  merchant  bought  38  gal.  of  wine  at  $2.15  per  gal. 
Did  he  gain  or  lose,  and  how  much,  by  selling  it  at  $5  per 
dekalitre? 

13.  Which  is  cheaper,  to  buy  cloth  at  $3  per  metre,  or  at 
$2.90  per  yard?     How  much  cheaper? 

14.  How  much  carpet  a  metre  in  width,  is  required  to  car- 
pet a  room  4.2  metres  long  and  3.8  metres  wide? 

15.  How  long  must  a  pile  of  wood  be,  so  that  it  may  con- 
tain 12  steres,  if  it  is  3.5  metres  wide  and  3  metres  high? 

16.  How  much  is  gained  by  selling  a  piece  of  silk  100 
metres  in  length,  at  $2.25  per  yard,  if  it  cost  $2  per  metre? 

17.  If  a  farm  contains  1400  ares,  what  will  be  its  value  at 
$2.50  per  are? 

18.  A  barrel  of  flour  contains  198  pounds.     Express  its 
weight  in  metric  units. 

19o  How  many  hectares  are  there  in  a  farm  that  is  1000 
metres  long  and  180  metres  broad?  What  is  its  value  at 
$250  per  hectare? 

20.  A  bin  is  5  metres  square  and  2.5  metres  high.     How 
many  hectolitres  of  wheat  will  it  hold  ? 

21.  A  room  is  5.2  metres  long,  4.5  metres  wide,  and  3.2 
metres  high.     What  will  be  the  cost  of  plastering  it  at  35 
cents  per  square  metre? 

22.  Which  is  more  profitable,  and  how  much  per  ton,  to 
sell  sugar  at  11  cents  per  Ib.  or  23  cents  per  kilo? 

23.  A  cask  holding  2  hectolitres  of  molasses  was  sold  at 
18f  cents  per  litre.     How  much  more  profitable  would  it  be 
to  sell  the  molasses  at  90  cents  per  gallon? 


ii 


PERCENTAGE 


i 

jj 


309.    1.  In  a  quantity  of  sugar,  4  Ib.  of  every  100  Ib.  were 
wasted.     What  part  of  it  was  wasted  ? 

2.  A  laborer  digs  potatoes  for  10  bushels  out  of  every  100. 
What  part  of  the  whole  does  he  get  ? 

3.  A  merchant  lost  $3  out  of  every  $100  worth  of  goods 
sold,  on  account  of  bad  debts.     What  part  of  his  sales  did 
he  lose? 

4.  Millers  take  1  bushel  out  of  every  10  bushels  which 
they  grind  for  customers,  as  pay  for  grinding.     How  many 
hundredths  do  they  take  ? 

5.  In  a  company  of  soldiers,  1  out  of  every  4  men  wras  killed. 
How  many  was  that  per  hundred,  or  per  cent,  f 

6.  In  a  school,  5  out  of  20  pupils  are  more  than  14  years 
old.    How  many  is  that  per  hundred  ?    How  many  per  cent.  ? 

7.  A  man  spent  $3  out  of  every  $4  earned.     How  many 
hundredths  of  his  money  did  he  spend?     What  per  cent.? 

8.  A  man  whose  income  was  $2500  annually,  saved  y1-^-, 
or  10  per  cent,  of  it.     How  many  dollars  did  he  save  ? 

9.  What  is  yfo-,  or  5  per  cent,  of  $800?     6  per  cent,  of 
$500? 

10.  What  is  2  per  cent,  of  $500?     4  per  cent,  of  $900? 
3  per  cent,  of  $500? 

11.  What  is  5  per  cent,  of  $800?     6  per  cent,  of  $900? 

12.  What  is  8  per  cent,  of  500  bushels?     10  per  cent,  of 
1000  pounds? 

(207) 


208 


PERCENTAGE. 


DEFINITIONS. 

310.  Per  Cent,  means  by  the  hundred. 

It  is  a  contraction  of  the  Latin  per  centum,  by  the  hundred. 

311.  The  Sign  offer  Cent,  is  %.     Thus,  S%  is  read 
8  per  cent. 

312.  Percentage  treats  of  computations  which  involve 
per  cent. 

313.  Since  per  cent,  is  a  number  of  hundredths,  it  is  usu- 
ally expressed  as  a  decimal.     It  may  also  be  expressed  as  a 
common  fraction.     Thus, 

2  per  cent,  is  written  2^,,     .02,  or  TJ  7. 

5  per  cent,  is  written  5^,,     .05,  or  Tf¥. 

47  per  cent,  is  written  47  ^>,     .47,  or  Ty7. 

135  per  cent,  is  written  135^,  1.35,  or  -}ff. 

12J  per  cent,  is  written  12^,  .12J,  or  f|$. 

f  per  cent,  is  written  f  ^>,  .00  f,  or  T§Q. 

31J  per  cent,  is  written  31J^>,  .31^,  or  fi$. 

314.  The  expressions  .12^,  .31^,  etc.,  may  also  be  written 
.125,  .3125,  etc.  ;  and  the  complex  fractional  forms  -J-g^,  yf  -Q, 
etc.  ,  may  be  expressed  as  simple  fractions  :  as,  -^j-,  -$$,  etc. 


Express  decimally,  and  in  the  smallest  terms  of  their  equiv- 
alent common  fractions,  the  following: 


1.     10%. 

9.       6i%. 

17.  18|% 

2.  12—%. 

10.     125%. 

18.  20|% 

3.     20%. 

11.       11%. 

19.      |% 

4.     25%. 

12.     33i%. 

20.      \% 

5.     30%. 

13.     16|%. 

21.     4i% 

6.     75%. 

14.  2121%. 

99         3  erf 
~l"Tf  /C 

7.  87i%. 

15.     31i%. 

23.  7T%% 

8.     3£%. 

16.     66|%. 

24.  37i% 

PERCENTAGE. 


209 


Express  in  hundredths  or  per  cent.,  ^  of  a  number;  \  of 
i>  iV;  "srV;  2T>  t>  T8Tr5  T^  A»  i5  f 5  i;  i»  i>  T>  irJ 

r  >    "o  »    T6  >     1  6  >    "5  0  >    7T- 

Problems  in  Percentage  involve  the  following  elements: 

315.  The  Base  is  the  number  of  which  the  per  cent,  is 
aken. 

316.  The  Hate  is  the  number  of  hundredths  taken. 

317.  The  Percentage  is  the  number  which  is  a  certain 
number  of  hundredths  of  the  base. 

318.  The   Amount  is  the  sum  of  the  base  and  per- 
centage. 

319.  The  Difference  is  the  base  less  the  percentage. 

In  the  formulas,  B.  represents  base ;  R.,  rate ;  P.,  percentage ;  A., 
amount;  and  D.,  difference. 

CASE  I. 

320.  To  find  the  Percentage  when  the  Base  and  Rate 
are  given. 


EXERCISES. 


1.  What  is 

2.  What  is 

3.  What  is 

4.  What  is 

5.  What  is 

6.  What  is 

7.  What  is 

8.  What  is 

9.  What  is 

10.  What  is 

11.  What  is 

14 


10  per  cent. 
5  per  cent. 
20  per  cent. 
2^  per  cent. 
15  per  cent. 

per  cent. 

per  cent. 
40  per  cent. 

4  per  cent. 

5  per  cent. 
25  per  cent. 


,  or  ^o,  of  $150? 
,  or -rfa,  of  $400? 
,  or  T2<£r ,  of  300  bu.  ? 
,  or  iff,  of  800  gal.? 
,  or  TVo,  of  $400? 
of  $600? 
of  160  men? 
of  200  tons? 
of  800  horses? 
of  700  pupils? 
of  124  yards? 


210  PERCENTAGE. 

12.  What  is  6J  per  cent,  of  $32.64? 

PROCESS. 

J4y_l_  —  $2.04  ANALYSIS. — Since    6J^>    of    any 

number  is  T6^,  or  j1^,  of  it,  6|^>  of 
Or,  $32.64  is  TV  of  $32.64,  which  is  $2.04. 


Since  6J^>  of  any  number  is  .06^ 
of  that  number,   6J$,   of  $32.64  is 

FORMTILA..  f\£»i       -C   c^or»  nA          1*1      •      tn»o  rk  j 

.06J  of  $32.64,  which  is  $2.04. 


RULE. — Multiply  the  base  by  the  rate. 


13.  Find  35%  of  $21.75. 

14.  Find  48%  of  $13.42. 


15.  Find  33|%  of  465  gal. 

16.  Find  37|%  of  816  mi. 


17.  A  farmer  who  had  a  flock  of  450  sheep,  sold  33-|% 
of  them.     How  many  had  he  left? 

18.  A  man  whose  salary  was  $2000  per  year,  spent  85% 
of  it.     How  much  did  he  save  annually? 

19.  A  farmer  sold  37^%  of  his  crop  of  816  bu.  of  wheat 
at  $1.56  per  bu.,  and  the  rest  at  $1.60.     How  much  did  he 
realize  from  the  sale  of  his  wheat  ? 

20.  If  a  merchant  makes  a  deduction  of  5%  from  a  bill 
of  $318.57,  how  much  must  be  paid  him? 

21.  A  man  bought  a  farm  for  $30000,  and  sold  it  for  a 
gain  of  25%.     How  much  did  he  get  for  it? 

22.  Mr.  Seymour  sold  $3000  worth  of  flour  at  a  loss  of 
12|-%.     How  much  did  he  realize  from  the  sale? 

23.  A  man  having  $40000,  invested  15%  in  bank  stock, 
27%  of  it  in  bonds  and  mortgages,  and  the  rest  in  a  flouring 
mill.     How  much  did  the  mill  cost? 

24.  Two  brothers  each  inherited  $18500.     The  elder  in- 
creased his  inheritance  8%  per  year  for  3  years.    The  younger 
lost  33-|%  of  his  in  the  same  time.    What  was  then  the  value 
of  the  inheritance  of  each  ? 


PERCENTAGE. 


211 


CASE  II. 

321.  To  find  the  Rate  when  the  Base  and  Percentage 
given. 

1.  If  a  man  earn  $100  and  spend  $50  of  it,  what  part  of 

,  does  he  spend  ?     How  many  hundredths  ?     How  many  per 
ent.? 

2.  In  a  piece  of  cloth  containing  36  yards,  9  yards  were 
imaged.    What  part  of  it  was  damaged?     How  many  hun- 

Iredths  of  it?     How  many  per  cent.? 

3.  When  I  spend  \  of  my  money,  how  many  hundredths 
of  it  do  I  spend  ?     How  many  per  cent.  ? 

4.  If  a  farmer  loses  %  of  his  crop  by  a  flood,  how  many 
hundredths  of  it  does  he  lose  ?    How  many  per  cent.  ? 

5.  If  a  merchant  sells  \  of  his  goods  annually,  how  many 
hundredths  does  he  sell  ?     How  many  per  cent.  ? 

6.  A  farmer  had  25  sheep,  and  10  of  them  died.     What 
part  of  his  sheep  died  ?     What  per  cent,  of  them  ? 

7.  What  part  of  $15  are  $3?    What  per  cent.? 

8.  What  part  of  12  bushels  are  6  bushels?    What  per 
cent.  ? 

9.  What  per  cent,  of  24  cows  are  8  cows?    What  per 
cent,  are  12  cows? 

10.  What  per  cent,  of  200  students  are  40  students?    Are 
60  students? 

11.  What  per  cent,  of  150  acres  are  30  acres?    Are  50 
acres?    Are  75  acres? 

12.  What  per  cent,  of  80  hours  are  16  hours?     Are  20 
hours?     Are  40  hours? 

13.  What  per  cent,  of  90  gallons  are  30  gallons?     Are  60 
allons?     Are  45  gallons? 

14.  If  a  man  who  earns  $60  per  month,  expends  $40  per 
aonth  for  necessary  expenses,  what  per  cent,  of  his  earnings 
loes  he  save? 


212  PERCENTAGE. 

15.  A  merchant  having  375  yards  of  cloth,  sold  150  yards 
of  it.     What  per  cent,  did  he  sell? 

PROCESS.  ANALYSIS. — 150  yards 

1 5  0  yd.  =  m  of  3  7  5  yd. ,  or  are2"«' or 

1  of  375  yd.,  or  40^  of  375  yd.       ^hu      40  hu 
Qr  dredths;     therefore    150 

yards    are   .40,   or   40%, 
150  yd.  -375yd. -.40,  or  40%        Of  375  yards.    Or, 

Since  the  percentage  is 

FORMULA.  a  product  of  the  base  by 

p  _._  £>  __  jj  the  rate,  if  we  divide  the 

percentage  by  the  base  we 

shall  obtain  the  rate.    Therefore  we  divide  150  by  375,  and  obtain  for 
a  quotient  .40,  or  40^. 

RULE. — Divide  the  percentage  by  the  base. 

16.  What  per  cent,  of  360  men  are  60  men? 

17.  What  per  cent,  of  840  men  are  360  men? 

18.  What  per  cent,  of  380  pages  are  120  pages? 

19.  What  per  cent,  of  45  hours  are  25  hours? 

20.  What  per  cent,  of  50  yards  are  27  yards? 

21.  What  per  cent,  of  36  pounds  are  24  pounds? 

22.  A  farmer  who  had  a  farm  of  540  acres,  sold  210  acres 
of  it.     What  per  cent,  of  it  did  he  sell? 

23.  A  man  whose  annual  income  is  $1800,  spends  $1600  of 
it.     What  per  cent,  of  it  does  he  spend?     What  per  cent,  of 
it  has  he  left? 

24.  A  grocer  sold  tea  for  $1  that  cost  him  $  .75.     What 
per  cent,  of  the  cost  did  he  gain? 

25.  What  per  cent,  of  30000  bushels  are  50  bushels? 

26.  What  per  cent,  of  the  cost  does  a  hatter  gain  by  sell- 
ing hats  at  $7  each,  that  cost  $5.50? 

27.  A  real-estate  agent  gets  $60  for  selling  my  house  for 
$4000.     What  %  of  the  sale  does  he  receive  for  his  services? 


PERCENTAGE. 


213 


28.  I  paid  SI 20  for  insuring  a  boat-load  of  wheat  valued 
at  $10000.    What  %  of  the  value  of  the  cargo  was  received 
for  insuring  it? 

29.  A  man  who  had  1000  acres  of  land,  gave  \  of  it  to  his 
eldest  son,  \  of  it  to  another,  and  the  remainder  he  divided 

Dually  between  his  3  daughters.     What  %  of  the  whole  did 
ch  receive? 

CASE  III. 

322.  To  find  the  Base  when  the  Percentage  and  Rate 
Eire  given. 

1.  A  man  spent  $15,  which  was  10%  or  y1-^-  of  all  the 
aoney  he  had.     How  much  money  had  he  ? 

2.  My  net  profit  from  an  investment  was  $800,  which  was 
25%  or  T2^5¥  of  the  amount  invested.     How  much  had  I  in- 
vested? 

3.  Of  what  sum  is  18  dollars  33^%,  or  fj$,  or  i? 

4.  Of  how  many  days  are  15  days  29%  ?    30  days  37^%  ? 

5.  Of  what  sum  is  25  dollars  62^%,  or  f|f,  or  f  ? 

6.  Of  what  number  is  120    6%  ?    150,  30%  ?    180,    60%  ? 

7.  Of  what  number  is    40  80%  ?      20,  60%  ?      30,  150%  ? 

8.  A  drover  lost  450  sheep,  which  was  75%  of  his  flock. 
How  many  sheep  had  he? 

ANALYSIS. — Since  75^  or  -££$  or  f  of  the 
number  is  450,  J  of  the  number  is  J  of  450 
or  150;  and  since  150  is  \  "of  the  number, 
the  whole  number  of  sheep  will  be  4  times 
150,  or  600.  Therefore  he  had  600  sheep. 
Or, 

Since  the  percentage  is  the  product  of 
the  base  by  the  rate,  if  the  percentage  is 
divided  by  the  rate  the  quotient  will  be  the 
base. 

Therefore  we  divide  450  by  .75. 


PROCESS. 

or     =  450 


Whole  =  600 

Or, 
450--.  75  =  600 

FORMULA. 


RULE. — Divide  the  percentage  by  tlie  rate. 


214  PERCENTAGE. 


Of  what  number  is 
9.    385   12$%  ?  , 

10.  245     10%? 

11.  125     15%? 

12.  7.15  33^%? 


Of  what  number  is 

13.  $53.25 

14.  27.5bu.    8%? 

15.  168  men  8%? 

16.  231  oxen  7% ? 


17.  A  farmer  sold  275  barrels  of  apples,  which  were  75% 
of  all  he  had.     How  many  had  he? 

18.  A  man  sold  25%  of  a  mill  for  $3750.     At  this  rate 
what  was  the  mill  worth  ? 

19.  A  man  who  owned  40%  of  a  foundry  sold  25%  of  his 
share  for  $10000.     What  was  the  value  of  the  foundry  ? 

20.  A  farmer  after  selling  110  A.  43  sq.  rd.  of  land  had 
90%  of  his  land  left.     How  much  land  had  he  at  first? 

21.  A  farm  cost  $3000.     One-third  of  this  sum  was  62$% 
of  what  the  house  and  barn  on  the  farm  cost.     What  was  the 
cost  of  the  house  and  barn? 

22.  A  man  indebted  to  me  paid  me  $80,  which  was  S$% 
of  $  the  amount  due.     How  much  did  he  still  owe? 

23.  A  merchant  sold  4500  bushels  of  wheat  at  $1.60  per 
bushel.     The  amount  received  was  90%   of  the  cost  of  the 
wheat.     How  much  did  it  cost? 

24.  Mr.  A.  sold  a  lot  for  $8000,  which  was  only  40%  of 
the  amount  he  paid  for  it.     How  much  did  he  pay  for  it  ? 

25.  A  man  pays  $600  a  year  rent;   75%  of  this  sum  is 
just  33$%  of  $  his  income.     What  is  his  income? 

26.  A  man  owning  $  of  a  vessel  sold  25%  of  his  share  for 
$3350.50.     At  that  rate  what  was  the  value  of  the  vessel? 

27.  The  amount  paid  by  insurance  companies  to  the  people 
of  St.  John,  New  Brunswick,  for  losses  caused  by  the  great 
fire  in  1877,  was  about  $7500000,  which  was  37$%  of  the 
estimated  loss.     What  was  the  estimated  loss  ? 

28.  25%  of  $  of  60  is  75%  of  $  of  what  number? 

29.  |  of  40%  of  100  is  5%  of  10  times  \  of  what  number? 


PEKCENTAGE.  215 


CASE  IV. 

323.  To  find  the  Base  when  the  Amount  and  Rate 
given. 

1.  A  gentleman  increased  his  collection  of  horses  by  an 
addition  of  ^  of  the  number,   and  then  he  had  15.     How 
many  had  he  before  he  made  the  addition  ? 

2.  A  coal  dealer  in  selling  coal  at  $6  a  ton  received  20% 
or  -J-  more  than  it  cost  him.     What  did  it  cost  him  ? 

3.  A  grocer  in  selling  sugar  at  $.11  a  pound  gains  10% 
or  -^  of  the  cost.     What  was  the  cost? 

4.  A  merchant  sold  cloth  at  an  advance  of  25%   on  the 
cost,  receiving  $1.25  per  yard  for  it.     What  was  the  cost? 

5.  A  man's  monthly  expenses  were  33^-%  more  during  1876 
than  during  1875.     During  1875  they  were  $120.    What  were 
they  in  1876? 

6.  A  certain  number  increased  by  20%   of  itself  is  36. 
What  is  the  number? 

7.  After  adding  to  a  number  37-^%  of  it,  the  sum  is  33. 
What  is  the  number? 

8.  What  number  increased  by  35%  of  itself  equals  540? 

PROCESS.  ANALYSIS. — Since  the  number  is  in- 

13. 5.  _  5  4  Q  creased  by  35^,  or  -f^  of  itself,  the  amount 

^1°  will  be  lT3o5Q-  or  ^  times  the  number; 

T™  7~  and  since  |J  J  of  the  number  ===  540,  Tfo 

The  number  =  of  u  =  _,_  of  ^  which  ig  4.  and  gince 

Or,  4  is  T^7  of  the  number,  the  number  will 

(1+35)  be  100  times  4,  which  is  400.     Or, 

A  number  increased  by  35^  of  itself 

equals  135/0  or  1.35  of  itself.     And  since 

1.35  times   the   number  equals   540,  the 

FORMULA.  number  may  be  found  by  dividing  540 

A~(1  +  E)=B.       by  1.35. 

RULE. — Divide  the  amount  by  1  -(-  the  rate. 


216  PEKCENTAGE. 

9.  What  number  increased  by    27%  of  itself  equals  508? 

10.  What  number  increased  by  33|-%  of  itself  equals  492  ? 

11.  What  number  increased  by  16f  %  of  itself  equals  329? 

12.  What  number  increased  by  62^-%  of  itself  equals  910? 

13.  A  man  owes  $15400,  which  is   10%   more  than  his 
property  is  worth.     What  is  the  value  of  his  property  ? 

14.  A  man  sold  a  horse  for  $345,  which  was  15%  more 
than  it  cost  him.     How  much  did  it  cost  ? 

15.  A  clerk's  salary  was  increased  30%,  and  now  it  is 
$1950.     What  was  it  before  the  increase? 

16.  A  man  expended  $3750  in  repairs  upon  his  house.    This 
sum  was  25%  more  than  -J-  the  cost  of  the  house.     How 
much  did  it  cost? 

17.  The  number  of  pupils  in  a  certain  school  during  1876 
was  872,  which  was  9%  more  than  the  number  in  attendance 
during  1875.     What  was  the  attendance  during  1875  ? 


CASE  V. 

324.  To  find  the  Base  when  the  Difference  and  Rate 
are  given* 

1.  A  gentleman  sold  25%  or  \  of  the  number  of  his  horses 
and  had  15  left.     How  many  had  he  ? 

2.  By  selling  coal  at  $6  per  ton  a  coal  dealer  lost  20%  or 
•J-  of  the  cost.     What  was  the  cost  ? 

3.  A  grocer  sold  sugar  at  9c.  per  pound,  and  lost  10%  or 
-^  of  the  cost.     What  did  it  cost? 

4.  A  merchant  sold  cloth  at  $  .75  per  yard,  thereby  losing 
25%  of  the  cost.     What  was  the  cost? 

5.  A  man's  monthly  expenses  are  33^%  less  this  year  than 
last  year.     This  year  they  are  $120 ;  what  were  they  last  year  ? 

6.  A  certain  number  diminished  by  20%  of  itself  is  36. 
What  is  the  number? 

7.  What  number  diminished  by  10%  of  itself  equals  45? 


PERCENTAGE.  217 

8.  After  subtracting  from  a  number  37|%  of  it,  the  re- 
minder is  25.     What  is  the  number? 

9.  What  number  diminished  by  27%  of  itself  equals  401.5? 

PROCESS.  ANALYSIS. — Since  the  number  is  de- 

73  —  401    5  creased  by  27^,,  or  T2^  of  itself,  the 

1i°  £    f-  remainder  will  be  •££$  of  the  number, 

which  equals  401.5;   T^F  of  it  equals 

The  number —  550          7V  of  401.5,  which  is  5.5;   and  since 
^  5.5  is  T^  of  the  number,  the  number 

'  will  be  100  times  5.5,  which  is  550. 

(1  — .27)  Or, 

401.  5-v-.  73  =  550  ^  number  diminished  by  27^   of 

itself,  equals  73$,,  or  .73,   of  itself; 
and  since   .73  of  the  number  equals 

FORMULA.  AM  ,      ,,  ,  .„    ,  , 

401.5,   the  number  will  be  equal  to 
D-±-(l  —  R)==B.          401.5 -f-  .73,  which  is  550. 

RULE. — Divide  the  difference  by  1  minus  the  rate. 

10.  What  number  diminished  by  36%  of  itself  equals  336? 

11.  What  number  diminished  by  40%  of  itself  equals  432? 

12.  What  number  diminished  by  55%  of  itself  equals  285? 

13.  What  number  diminished  by  28%  of  itself  equals  307? 

14.  A  clerk,  after  paying  out  75%  of  his  salary,  had  $450 
left.     What- was  his  salary? 

15.  A  farmer,  after  selling  30%  of  his  wheat,  found  that 
he  had  350  bushels  left.     How  much  had  he  at  first? 

16.  A  man  sold  some  land  for  30%  less  than  he  asked  for 
it,  getting  $29.24  per  acre.     What  was  his  asking  price? 

17.  A  regiment  losing  15%  of  its  men,  had  527  left.     How 
many  had  it  at  first? 

18.  A  speculator  lost  10%  of  his  money  during  the  year 
1875  and  10%  of  the  remainder  during  1876.     He  then  had 
$40500  left.     How  much  had  he  at  first? 

19.  A  merchant's  profit  in  1876  was  $10318,  which  was 
23%  less  than  in  1875.     What  was  his  profit  in  1875? 


74i  --'-.,-.'   .....---::  -^j^^^ 

xiltffr,  ;mimiiimimninj  JJMiMIA^ 


325.  1.  When  a  sum  equal  to  5%   of  the  amount  of 
money  lent  is  paid  for  the   use   of  it  for  one  year,   how 
much  will  be  paid  for  the  use  of  $100  for  1  year?     For  2 
years? 

2.  When  the  allowance  for  the  use  of  money  is  6%  per 
year,  what  is  the  allowance  for  the  use  of  $100  for  1  year? 
For  2  years?     For  3  years?     For  3^  years? 

3.  When  the  sum  paid  for  the  use  of  money  is  8%  per 
year,  what  must  be  paid  per  year  for  $50?     For  $500? 

4.  When  the  sum  paid  for  the  use  of  money  is  12%  yearly, 
what  must  be  paid  for  the  use  of  $100  for  1  year? 

5.  When  the  allowance  for  the  use  of  money  is  8%  per 
year,  what  must  be  paid  for  the  use  of  $100  for  6  months? 
For  1  month?    For  -J-  month?    For  %  month?    For  £  month? 
For  10  days?     For  20  days? 

6.  When  6%  is  paid  per  year  for  the  use  of  money,  how 
much  will  $500  amount  to  in  2  years?     In  3  years? 

7.  When  $500  is  loaned  for  1^  years  at  8%  per  year,  what 
will  be  the  amount? 

DEFINITIONS. 

326.  Interest  is  the  sum  paid  for  the  use  of  money. 

327.  The  Principal  is  the  sum  for  the  use  of  which 

interest  is  paid. 

(218) 


INTEREST. 


219 


328.  The  Amount  is  the  sum  of  the  principal  and  in- 
terest. 

329.  The  Hate  of  Interest  is  the  annual  rate  per 
cent. 

330.  Legal  Interest  is  interest  according  to  rate  fixed 
by  law. 

331.  Usury  is  interest  computed  at  a  higher  rate  than 
the  law  allows. 

332.  A  Note,  or  Promissory  Note,  is  a  written 
promise  to  pay  a  sum  of  money  at  a  given  time. 

333.  PRINCIPLE. — The  interest  is  equal  to  the  product  of  the 
principal,  rate,  and  time  expressed  as  years. 

334.  When  the  rate  per  cent,  is  not  specified  in  notes, 
accounts,  etc.,  the  legal  rate  is  always  understood. 

On  debts  due  the  United  States  the  rate  is  6^ . 

The  following  table  contains  the  rates  of  interest  in  the 
United  States.  The  first  column  gives  the  legal  rate,  the 
second  the  rate  that  may  be  collected  if  agreed  to  in  writing. 


NAME   OF  STATE. 

R; 
PER 

TE 

CENT. 

NAME    OF  STATE. 

RJ 
PER 

TE 

CENT. 

8 

Mississippi  

6 

10 

Arkansas  

6 

10 

6 

10 

10 

Any. 

10 

Any. 

10 

Any 

6 

Con  necticut  

7 

New  Jersey 

7 

10 

Any. 

New  Mexico  

6 

Any. 

Dakota.       ...       ...                 . 

7 

12 

New  York.     .     . 

7 

Delaware  

6 

North  Carolina  

6 

8 

District  of  Columbia 

g 

10 

10 

12 

8 

Any 

Nevada 

10 

7 

12 

Ohio  

6 

*Y- 

Idaho  

10 

24 

10 

12 

Illinois  

6 

10 

Pennsylvania  

6 

Indiana  

Indian  Territory  

6 

10 

Rhode  Island  
South  Carolina  

6 

7 

Any. 
Any 

Iowa. 

g 

10 

• 

10 

Kansas  

7 

12 

Texas.                

8 

Kentucky    .     . 

6 

10 

Utah 

10 

Any 

Louisiana  

5 

g 

g 

Maine  

6 

Any 

Virginia  

g 

Maryland  

6 

West  Virginia. 

g 

Massachusetts  

6 

Any 

Washington  Territory  

10 

Any. 

Michigan    

7 

10 

7 

10 

Minnesota  

7 

12 

Wvominc 

12 

220  PEKCENTAGE. 


TO  COMPUTE  INTEREST. 

335.   1.  What  is  the  interest  of  $200  for  1  year  at  6^  ? 

2.  What  is  the  interest  of  $200  for  2  years     at  7^  ? 

3.  What  is  the  interest  of  $300  for  3  years     at  5^  ? 

4.  What  is  the  interest  of  $400  for  1£  years   at  8  ft  ? 

5.  What  is  the  interest  of  $400  for  3  months  at  6  ft  ? 

6.  What  is  the  interest  of  $600  for  1  month   at  6^  ? 

7.  What  is  the  interest  of  $600  for  10  days    at  6^  ? 

8.  What  is  the  interest  of  $500  for  15  days    at  8^  ? 

9.  What  will  be  the  amount  of  $100  for  2    years  at  6  ft  ? 

10.  What  will  be  the  amount  of  $200  for  3    years  at  4^  ? 

11.  What  will  be  the  amount  of  $300  for  2£  years  at  6^  ? 

12.  What  will  be  the  amount  of  $150  loaned  for  1^  years 
at  5^? 

13.  Find  the  interest  of  $234.27  for  2  yr.  7  mo.  12  da.  at  6^  ? 

PROCESS.  ANALYSIS. — Since  the   in- 

$  2  3  4    27  terest   for   1    year    is  6^   of 

^  ^  the  principal,  we  find  .06  of 

$234.27,  which    is   $14.0562; 

$14.0562  Int.  for  1  yr.  and  since  $14.0562  is  the  in- 

2  terest  for  1  year,  the  interest 

$28.1124  Int.  for  2  yr.  for  2  7ears  wil1  be  twice  that 

Q    a  a  n  a  T  sum,  which  is  $28.1124.    The 

8.6676  Int.  for  7  2-5  mo. 

interest  for  1  month   is  one- 

$36.78         Int.  2  yr.  7  mo.  12  da.       twelfth  of  the  interest  for  1 

^  year,  or  $1.1713  ;  and  the  in- 

'  terest  for  7  mo.  and  12  days, 

$  2  3  4    2  7  or  7f  nio.,  is  7|  times  $1.1713, 

or  $8.6676.   This  added  to  the 
interest  for  2  years,  gives  the 


1  2  )$14.Q562  Int.  for  1  yr.  interest  for  2  years,  7  months, 


1.1713  Int.  for  1  mo.                   12  da^s-     Or> 

o  -.     A  We  may  find  the  interest 

-  '.  —  for  1  year  as  before,  and  then 

$36.78  Int.  2  yr.  7  mo.  12  da.       for  1  month.     We  then  mul- 


INTEREST.  221 

tiply  the  interest  for  1  month  by  the  number  of  months  and  fractions 
of  a  month.  Thus,  in  2  years,  7  months,  there  are  31  months,  and  in 
12  days  there  are  Jf  or  T%  of  a  month. 

Therefore,  the  entire  interest  may  be  found  by  multiplying  $1.1713 
by  31.4,  which  is  $36.78. 

Since  there  are  30  days  in  a  month,  one-third  of  the  number  of 
days  will  be  tenths  of  a  month. 

EULE. — I.  Find  the  interest  for  1  year  and  multiply  this  by 
the  time  expressed  as  years  and  fractions  of  a  year.  Or, 

II.  Find  the  interest  for  1  month  and  multiply  this  by  the  time 
expressed  as  months  and  fractions  of  a  month. 

14.  What  is  the  interest  of  $25.16  for  1  yr.  6  mo.  at  6^  ? 

15.  What  is  the  interest  of  $36.24  for  2  yr.  4  mo.  at  1%  ? 

16.  What  is  the  interest  of  $48.20  for  2  yr.  4  mo.  at  8^  ? 

17.  What  is  the  interest  of  $2000   for  3  yr.  7  mo.  at  9^  ? 

18.  Find  the  amount  of  $590.50  for  3  yr.  6  mo.  at    1%. 

19.  Find  the  amount  of  $640.82  for  2  yr.  7  mo.  at    8^. 
(  20.  Find  the  amount  of  $725.83  for  3  yr.  6  mo.  at 

21.  Find  the  amount  of  $618.24  for  2  yr.  5  mo.  at 

22.  Find  the  amount  of  $312.29  for  3  yr.  5  mo.  at 

23.  Find  the  interest  of  $718.24  for  5  mo.  10  da.  at 

24.  Find  the  interest  of  $127.46  for  3  mo.  15  da.  at 

25.  Find  the  interest  of  $364.18  for  2  mo.  12  da.  at 

26.  Find  the  interest  of  $318.29  for  9  mo.  10  da.  at 

27.  Find  the  interest  of  $312.24  for  2  mo.  20  da.  at 

28.  Find  the  interest  of  $1614.25  for  20  da.  at      7 

29.  Find  the  interest  of  $1318.29  for  24  da.  at 

30.  Find  the  interest  of  $4684.68  for  11  da.  at 

31.  If  you  lend  $500,  how  much  will  be  due  you  in  3  yr. 
6  mo.  21  da.,  interest  at  7  ft  ? 

32.  What  is  the  interest  on  $784.25  from  Aug.  7,  1874,  to 
July  19,  1877,  at  8^  ?     What  is  the  amount? 

33.  How  much  interest  is  due  on  $500,  that  has  been  loaned 
at  interest  since  Jan.  1,  1876? 


222  PEBCENTAGE. 


OTHER  METHODS. 

ALIQUOT  PARTS. 

336.    1.  What  is  the  interest  and  amount  of  $520.32  for 
2  yr.  5  mo.  15  da.  at  If0  ? 

PROCESS.  ANALYSIS. — Since    the   in- 

$520.32  terest  is  7  ft  of  the  principal, 

m  07  we  find  .07  of  $520.32,  which 

is  $36.4224,  the  interest  for  1 
$36.4224lnt.forlyr.  y^      Twice    ^^    ^ 

the  interest  for  2  years,  which 


$72.8448  Int.  for  2  yr.  is  $72.8448.     One-third  of  the 

12.1408  Int.  for  4  mo.  interest  for  1  year  is  $12.1408, 

3.0352lnt.forlmo.  the    interest    f°r    4    m°nths' 

^    e  .  -  n  One-fourth  of  the  interest  for 

1 .  5 1  7  O  Int.  for  15  da.  A  ,       .     ^  nor0    ,,       . 

4  months  is  $3.0352,  the  in- 


$    89.5384  Int.  for  2  yr.  5  mo.  15  da,       terest  for  1  month.     One-half 
$520.32         Principal.  the   interest  for   1   month    is 

ac AH    Q£  $1.5176,    the   interest   for   15 

$609.86         Amount.  '  ... 

days.      The    sum     of     these 

amounts  is  the  interest  for  2  years  5  months  15  days,  which  is  $89.5384. 
This  sum,  added  to  the  principal,  gives  the  amount. 

Solve  the  following  by  aliquot  parts: 

2.  What  is  the  interest  of  $324.22  for  3  yr.  4  mo.  at  6^  ? 

3.  What  is  the  interest  of  $218.90  for  2  yr.  7  mo.  at  7^  ? 

4.  What  is  the  interest  of    $36.48  for  2  yr.  5  mo.   15  da. 
at  6^? 

5.  What  is  the  interest  of    $40.28  for  1  yr.  7  mo.  20  da. 
at  6^? 

6.  What  is  the  interest  of    $56.24  for  2  yr.  5  mo.  18  da. 
at  7^? 

7.  What  is  the  interest  of    $24  96  for  3  yr.  1  mo.     6  da. 
at  8? 


INTEREST.  223 

8.  What  is  the  interest  of  $48.72  for  2  yr.  2  mo.  16  da. 

9.  What  is  the  interest  of  $36.18  for  2  yr.  5  mo.  21  da. 

10.  What  is  the  interest  of  $20.25  for  3  yr.  1  mo.  16  da. 

11.  What  is  the  interest  of  $30.24  for  2  yr.  8  mo.  15  da. 

t  8^? 

SIX  PER  CENT.  METHOD. 

337.  The  interest  on  $1,  at  6^  per  annum, 

For  12  months,  is 06 

For    2  months,  is 01 

For  1  month  (30  days),  is.  .  .  .005 
For  6  days  (|  month),  is.  .  .  .001 
For  1  day,  is 


1.  What  is  the  interest  of  $125  for  2  yr.  3  mo.  16  da.  at  6^  ? 

PROCESS.  ANALYSIS. — Int.  of  $1  for    2  yr.  at  6^,  is  .12. 

a  I  2  5  Ink  °^  $1  f°r    3  mo.  at  6^>  is  .015. 

.,  0  7  4  Int.  of  $1  for  16  da.  at  6f0  is  .002$. 

The  sum  of  these,  .137f ,  is  the  interest  of  $1  for 


$17.208          the  given  time  at  the  given  rate,  and  since  the 
interest  of  $1  is  .137£,  the  interest  of  $125  will  be 
times  that  sum,  which  is  $17.208. 

1.  When  it  is  required  to  find  the  interest  at  any  other  rate  than 
6^>,  first  find  it  at  6^,,  then  increase  or  decrease  this  result  by  such  a 
part  of  it  as  the  given  rate  is  greater  or  less  than  6^.     Thus,  if  the 
rate  is  7^,  increase  the  interest  at  6^  by  1  of  it;  if  the  rate  is  5^. 
decrease  it  by  £  of  it ;  if  the  rate  is  8^,  increase  it  by  -f ,  or  |  of  it  ; 
if  the  rate  is  9^,  increase  it  by  J  of  it,  etc. 

2.  Exact  or  Accurate  interest  requires  that  the  year  should  be 
considered  365  days,  for  a  common  year,  and  366  days  for  a  leap  year, 
instead  of  the  ordinary  method  of  considering  12  months  of  30  days 
each,  or  360  days  a  year. 


224  PERCENTAGE. 

Find  the  interest  on  the  following : 

2.  On  a  note  for  $185.26  for  1  yr.  4  mo.  13  da.  at 

3.  On  a  note  for  $368.18  for  3  yr.  5  jno.  22  da.  at 

4.  On  a  note  for  $284.25  for  2  yr.  7  mo.  18  da.  at 

5.  On  a  note  for  $183.17  for  1  yr.  8  mo.  17  da.  at 

6.  On  a  note  for  $215.25  for  3  yr.  2  mo.  18  da.  at 

7.  On  a  note  for  $204.37  for  2  yr.  5  mo.  15  dar  at 

8.  On  a  note  for  $186.15  for  3  yr.  7  mo.  23  da.  at 

9.  On  a  note  for  $315.30  for  1  yr.  9  mo.  27  da.  at 

10.  On  a  note  for  $379.15  for  1  yr.  8  mo.  11  da.  at 

11.  On  a  note  for  $685.31  for  4  yr.  1  mo.  15  da.  at 

12.  On  a  note  for  $516.28  for  3  yr.  6  mo.  28  da.  at 

13.  On  a  note  for  $423.15  for  2  yr.  7  mo.  10  da.  at 

14.  On  a  note  for  $304.27  for  1  yr.  3  mo.  21  da.  at 

15.  On  a  note  for  $516.24  for  2  yr.  1  mo.  13  da.  at  12%  ? 

Find  the  amount  of  the  following  notes  when  due : 

16.  $150.15.  CINCINNATI,  0.,  Jan.  31, 1877. 

Three  months  after  date,  for  value  received,  I  promise  to  pay 
John  T.  Jones,  or  order,  One  Hundred  Fifty  -£fa  Dollars,  with 
interest  at  6fi.  CHARLES  C.  THOMSON. 

17.  $328.35.  ST.  PAUL,  MINN.,  Oct.  1, 1877. 

On  the  15th  day  of  January,  1878,  for  value  received,  I  promise 
'  to  pay  "to  S.  E.  Hoyt,    or  order,  Three  Hundred  Twenty-Eight 
T3^  Dollars,  with  interest  at  8%.  J    W    KAY 

l&.: $31$.75*.  BUFFALO,  N.  Y.,  May  3,  1876. 

For  value  received,  on  demand  I  promise  to  pay  to  J.  C.  Coe, 
'of  order,  Three  Hundred/ Fifteen  -$fo  Dollars,  with  interest. 

HENRY  B.  EOBESON. 
» 

Paid,  June  5th,  1877.     How  much  was  due? 


INTEKEST. 


225 


COMPOUND  INTEREST. 

338.  Compound  Interest  is  interest  upon  the  prin- 
cipal and  its  unpaid  interest,  combined  at  regular  intervals. 

It  is  usually  compounded  annually,  semi-annually,  or  quarterly. 
Unless  some  other  condition  is  mentioned  in  the  written  obligation, 
the  interest  is  understood  to  be  compounded  annually. 


WRITTEN    EXERCISES. 

1.  Find  the  compound  interest  of  $250  for  2  yr.  3  mo.  at  ( 


$250 
15 


PROCESS. 
Prin.  for  1st  yr. 
Int.  for  1st  yr. 
Prin.  for  2d  yr. 


$265 

15.90  Int.  for2dyr. 
$280.90  Prin.  for  3d  yr. 

4.21  Int.  for  3  mo. 
$285.11  Amount  for  2  yr.  3  mo. 
250.00  First  Principal. 


$35.11  Comp.  Int.  for  2  yr.  3  mo. 
inal  principal,  and  obtain  $35.11,  the  compound  interest  required. 


ANALYSIS. — Since  the  in- 
terest is  compounded  an- 
nually, we  first  find  the  inter- 
est of  $250  for  1  yr.  We  add 
this  interest  to  the  principal, 
and  compute  the  interest  on 
this  amount  for  another  year. 
We  add  this  interest  to  the 
principal  as  before,  and  com- 
pute interest  on  this  amount 
for  3  months,  which  we  add 
to  the  principal.  From  this 
amount  we  subtract  the  orig- 


RULE. — Find  the  interest  of  the  principal  for  the  first  period 
of  time  at  the  end  of  which  interest  is  due. 

Add  this  interest  to  the  principal,  and  compute  the  interest 
upon  this  amount  for  the  next  period,  and  so  continue. 

Subtract  the  given  principal  from  the  last  amount,  and  the. re- 
mainder will  be  the  compound  interest. 

1.  If  the  interest  is  compounded  semi-annually,  the  rate  is  consid- 
ered as  one-half  the  annual  rate;  if  quarterly,  one-fourth,  etc. 


226  PERCENTAGE. 

2.  When  the  time  consists  of  years,  months,  and  days,  find  the 
compound  interest  for  the  greatest  number  of  entire  periods,  and  to 
this  add  the  simple  interest  upon  the  amount  for  the  rest  of  the  time. 

2.  Compute  the  compound  interest  on  $315  for  2  yr.  6 
mo.  at  6^. 

3.  Find  the  amount  of  $324.18  for  3  yr.  5  mo.  at  7 
compound  interest. 

4.  What  is  the  compound  interest  on  $525.75  for  3  yr. 
4  mo.  at  6^  ? 

Computations  in  compound  interest  may  be  shortened  very 
much  by  the  use  of  the  table  on  the  following  page. 

5.  What  is  the  compound  interest  of  $325.10  for  3  yr. 
2  mo.  at  &%  ? 

ANALYSIS. — By  referring  to  the  table,  the  amount  of  $1  for  3  yr.  is 
found  to  be  $1.191016.  Computing  interest  on  this  sum  for  the  remain- 
ing 2  mo.,  the  amount  is  $1.208881.  Since  the  amount  of  $1  for  the 
given  time  is  $1.208881,  the  amount  for  $325.10  will  be  325.10  times 
that  sum.  If  from  this  product  the  principal  is  subtracted,  the  re- 
mainder is  the  compound  interest. 

6.  Find  the  compound  interest  of  $600.50  for  3  yr.  7  mo. 
at  6^. 

7.  Find  the  compound  interest  of  $318.25  for  2  yr.  4  mo. 
at  lfc. 

8.  Find  the  compound  interest  of  $412.08  for  3  yr.  2  mo. 
10  da.  at  6^. 

9.  Find  the  compound  interest  of  $310.24  for  2  yr.  5  mo. 
15  da.  at  8^. 

10.  What  is  the  difference  between  the  simple  interest  on 
$328  for  2  yr.  7  mo.  at  7^,  and  the  compound  interest  on 
same  sum  for  the  same  time  at  6^  ? 

11.  If  I  deposit  $300  in  a  savings  bank  which  compounds 
at  6^  semi-annually,  how  much  will  be  due  me  in  3^  years  ? 


INTEREST. 


227 


COMPOUND    INTEREST    TABLE, 

Showing  the  amount  of  $1,  at  various  rates,  compound  int.  from  1  to  20  years. 


Yrs. 

1%  per  cent. 

3  per  cent. 

3>£  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

~T~ 

1.025000 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159693 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

7 

1.188686 

1.229874 

1.272279 

1.315932 

1.407100 

1.503630 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

12 

1.344889 

1.425761 

1.511069 

1.601032 

1.795856 

2.012197 

13 

1.378511 

1.468534 

1.563956 

1.665074 

1.885649 

2.132928 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.979932 

2.260904 

15 

1.448298 

1.557967 

1.675349 

1.800944 

2.078928 

2.396558 

16 

1.484506 

1.604706 

1.733986 

1.872981 

2.182875 

2.540352 

17 

1.521618 

1.652848 

1.794676 

1.947901 

2.292018 

2.692773 

18 

1.559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

19 

1.598650 

1.753506 

1.922501 

2.106849 

2.526950 

3.025600 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

Yrs. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  cent 

11  per  cent. 

12  per  cent. 

1 

1.070000 

1.080000 

1.090000 

1.100000 

1.110000 

1.120000 

2 

1.144900 

1.166400 

1.188100 

1.210000 

1.232100 

1.254400 

3 

1.225043 

1.269712 

1.295029 

1.331000 

1.367631 

1.404908 

4 

1.310796 

1.360489 

1.411582 

1.464100 

1.518070 

1.573519 

5 

1.402552 

1.469328 

1.538624 

1.610510 

1.685058 

1.762342 

6 

1.500730 

1.586874 

1.677100 

1.771561 

1.870414 

1.973822 

7 

1.605781 

1.713824 

1.828039 

1.948717 

2.076160 

2.210681 

8 

1.718186 

1.850930 

1.992563 

2.143589 

2.304537 

2.475963 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.558036 

2.773078 

10 

1.967151 

2.158925 

2.367364 

2.593742 

2.839420 

3.105848 

11 

2.104852 

2.331634 

2.580426 

2.853117 

3.151757 

3.478549 

12 

2.252192 

2.518170 

2.812665 

3.138428 

3.498450 

3.895975 

13 

2.409845 

2.719624 

3.065805 

3.452271 

3.883279 

4.363492 

14 

2.578534 

2.937194 

3.341727 

3.797498 

4.310440 

4.887111 

15 

2.759031 

3.172169 

3.642482 

4.177248 

4.784588 

5.473565 

16 

2.952164 

3.425943 

3.970306 

4.594973 

5.310893 

6.130392 

17 

3.158815 

3.700018 

4.327633 

5.054470 

5.895091 

6.866040 

18 

3.379932 

3.996019 

4.717120 

5.559917 

6.543551 

7.689964 

19 

3.616527 

4.315701 

5.141661 

6.115909 

7.263342 

8.612760 

20 

3.869684 

4.660957 

5.604411 

6.727500 

8.062309 

9.646291 

228  PERCENTAGE. 


ANNUAL  INTEREST. 

339.  Anmial  Interest  is  simple  interest  on  the  prin- 
cipal and  upon  any  interest  overdue,  when  the  contract  con- 
tains the  words,  "with  annual  interest,"  or,  "with  interest 
payable  annually." 

Annual  interest  is  not  considered  legal  in  some  States. 

1.  Find  the  amount  of  $3500  for  4  yr.  6  mo.,  with  inter- 
est payable  annually  at  6^5  < 

PROCESS.  ANALYSIS. — Since 

Int.  of  $3500  for  4J  yr.  ==   $945  annual  interest  is  sim- 

Int.  of  $210    for  8    yr.=±=   $100.80       pl?    !ntf est  °n    the 

principal    and     upon 

Annual  Interest      —$1045.80        any  over-due  interest, 
83500  +  81045.80  =  84545.80,  Amt.      we  first  find  the  inter' 

est  upon  the  principal, 

which  is  $945,  and  then  upon  the  interest  due.  The  interest  for  each 
year  is  $210.  The  interest  for  the  first  year  remained  unpaid  for  3J 
years ;  that  for  the  second  year,  2J  years;  that  for  the  third  year,  1| 
years;  and  that  for  the  fourth  year,  for  \  year;  therefore  the  annual 
interest,  $210,  drew  interest  for  3J  +  2£+l£+J  years,  or,  8  years, 
which  is  $100.80.  This  sum,  added  to  the  simple  interest,  $945  — 
$1045.80,  the  annual  interest.  This  sum  added  to  the  principal, 
$3500  =  $4545.80,  the  amount  due. 

RULE. — Compute  the  interest  on  the  principal  for  the  entire 
time,  and  on  each  year's  interest  from  the  time  it  was  due  up  to 
the  end  of  the  period. 

The  sum  of  these  interests  will  be  the  annual  interest. 

2.  How  much  -is  due  upon  a  note  of  $350  which  has  run 
4  years,  interest  at  8^,  payable  annually? 

3.  How  much  wras  due  April  15,  1877,  on  a  note  for  $750, 
dated  Jan.  1,  1873,  with  interest  at  6^,  payable  annually? 


INTEREST.  229 


PARTIAL   PAYMENTS. 

340.  A  Partial  Payment  is  a  payment  in  part  of  a 
note  or  other  obligation. 

341.  An  Indorsement  is  the  statement  of  the  amount 
of  a  payment  and  the  time  when  it  was  made.     It  is  written 
on  the  back  of  the  note  or  other  written  obligation. 

34:2.  Business  men  often  settle  notes  and  accounts  running 
for  one  year  or  less  by  what  is  known  as  the  Mercantile 
Rule. 

MERCANTILE  RULE. — Find  the  amount  of  the  principal  at  the 
time  of  settlement. 

Find  the  amount  of  each  payment  from  the  time  it  was  made 
until  the  time  of  settlement,  and  from  the  amount  of  the  principal 
subtract  the  amounts  of  the  payments. 

1.  A  note  for  8850,  on  demand  with  interest  at  Ifo  dated 
Jan.  1,  1876,  was  indorsed  as  follows:  April  10,  1876,  $200; 
Sept.  15,  1876,  $255.     How  much  was  due  Nov.  15,  1876? 

2.  What  is  the  balance  due  at  the  end  of  a  year  on  a  note 
for  $1800,  dated  May  15,  1875,  on  which  the  following  pay- 
ments had  been  made:  Sept.  20,  1875,  $300;  Jan.  18,  1876, 
$200;  April  20,  1876,  $1000;  when  the  rate  is  1%  ? 

3.  $585.25.  BUFFALO,  N.  Y.,  March  3,  1876. 
Eight  months  after  date,  for  value  received  I  promise  to  pay  to 

the  order  of  E.  8.  Farran,  Five  Hundred  Eighty-five  T2^  Dollars, 
with  interest  at  1%.  H.  S.  LAUPHIEE. 

This  note  was  indorsed  as  follows:  June  8,  1876,  $325; 
Aug.  4,  1876,  $84.30;  Sept.  2,  1876,  $100.  What  was  due 
on  the  note  at  maturity? 


230  PERCENTAGE. 

343.  Most  of  the  States  have  adopted  the  United  States 
Hule  for  computing  the  amount  due  upon  any  obligation 
where  partial  payments  are  made,  based  upon  the  following 
principle. 

PRINCIPLE. — The  indebtedness  should  be  computed  whenever  a 
payment  is  made,  but  the  principal  must  not  be  increased  by  the 
addition  of  interest. 

WRITTEN    EXERCISES. 

1.  A  note  was  given,  Jan.  1,  1870,  for  $700.  The  fol- 
lowing payments  were  indorsed  upon  it:  May  6,  1870,  $85; 
July  1, 1871,  $40;  Aug.  20,  1871,  $100;  Jan.  10,  1873,  $350. 
How  much  was  due  Sept.  30,  1874,  interest  at  6%? 

PROCESS. 

Principal $700.00 

Int.  to  May  6,  1870,— 4  mo.  5  da.         .        .        .        .        .  _14^>? 

Amount 714.58 

First  payment 85.00 

New  Principal 629,58 

Int.  from  May  6,  1870,  to  July  1,  1871,—!  yr.  1  mo. 

25  da 43.55 

Second  payment,  less  than  interest  due         .         .       $40.00 
Int.  on  $629.70  from  July  1,  1871,  to  Aug.  20, 1871 — 

1  mo.  19  da 5.14 

Amount 678.27 

Third  payment  to  be  added  to  second  .         .         .          $100  J40.00 

New  Principal .  538.27 

Int.  from  Aug.  20, 1871,  to  Jan.  10, 1873,—!  yr.  4  mo. 

20  da.      ...                 44.85 

Amount     .         .         .         .  •      .         .         .  583.12 

Fourth  payment 350.00 

New  Principal .  233.12 

Int.  from  Jan.  10,  1873,  to  Sept.  30, 1874  — 1  yr.  8  mo.  20  da.  24.08 

Amount  due,  Sept.  30,  1874     .        .                 .  $257.20 


PARTIAL  PAYMENTS.  231 

UNITED  STATES  KULE. — Find  the  amount  of  the  principal 
tc  a  time  when  a  payment,  or  the  sum  of  the  payments,  equals 
or  exceeds  the  interest  due,  and  from  this  amount  subtract  such 
payment  or  payments.  With  the  remainder  as  a  new  principal, 
proceed  as  before. 

2.  A  note  for  $2500,  dated  July  10,  1871,  bore  the  fol- 
lowing indorsements:    Sept.  15,  1871,  $150;  Nov.  12,  1871, 
$300;    Dec.  1,  1871,  $100;    April  3,  1872,  $325;    May  15, 

1872,  $275;    Sept.  20,  1872,   $1000.     How  much  was  due 
Jan.  1,  1873,  the  rate  of  interest  being  6%? 

3.  How  much  was  due  at  maturity  on  a  note  for  $2150, 
dated  Sept.  20,  1873,  to  run  2  years  6  months,  on  which  the 
following  payments  were  indorsed:  Dec.  15,  1873,  $75;  Feb. 
4,  1874,  $200;    April  3,  1874,  $150;    July  1,  1874,  $500; 
Dec.  16,  1874,  $1000,  the  rate  of  interest  being  8%? 

4.  A  note  for  $6725,  dated  Feb.  10, 1875,  had  the  following 
indorsements:   May  5,  1875,  $275;  Aug.  15,  1875,  $50;  Nov. 
12,  1875,  $1000;   Jan.  3,   1876,  $184.25;    Sept.   13,  1876, 
$84.10;  Dec.  23,  1876,  $1000.     How  much  was  due  Feb.  10, 
1877,  interest  at  6%  ? 

5.  A  bond  was  given  April  4,  1870,  for  $5825,  with  inter- 
est at  8%.     The  following  payments  were  indorsed  upon  it: 
May  15,  1871,  $728.50;  April  8,  1872,  $1000;  Dec.  12, 1872, 
$125;   July  9,  1873,  $980;   June   12,  1874,  $1000.      How 
much  remained  due  upon  the  bond  April  4,  1875? 

6.  Sept.  25,  1872,  James  Hanna  gave  his  note  for  $895.75 
with  interest  at  10%.     He  paid  on  it  as  follows:    Jan.  10, 

1873,  $25;  Oct.  12,  1873,  $200;  Jan.  18,  1874,  $75;  March 
25,  1874,  $187.50;  Jan.  1,  1875,  $375.     How  much  was  due 
when  he  paid  the  note,  Nov.  15,  1875? 

7.  Required,  the  balance  due  on  a  note  dated  Jan.  1,  1875, 
for  $580  at  5^,  to  run  2  years,  on  which  a  payment  of 
was  made  every  3  months. 


232  PERCENTAGE. 

8.  A  note  for  $10000,  with  interest,  dated  Milwaukee,  Wis., 
Dec.  12,  1875,  was  indorsed  as  follows:  Feb.  23,  1876,  $750; 
July  17,  1876,  $108.25 ;  Nov.  23,  1876,  $3000 ;  Jan.  18, 1877, 
$4000.    How  much  was  due  May  12,  1877,  interest  at  8^  ? 

9.  Required,  the  balance  due  July  8,  1876,  on  a  note  for 
$3124.75,  at  8^  interest,  dated  Feb.  15,  1874,  on  which  a 
payment  was  made  Dec.  23, 1874,  of  $985 ;  another  of  $875.35, 
Feb.  15,  1875;  another  of  $1025,  Feb.  20,  1876. 

10.  $1885.75.  SCHENECTADY,  N.  Y.,  Feb.  10, 1874. 
Two  years  after  date,  for  value  received  I  promise  to  pay 

W.  W.   Heilbronner  or  order.  One  Thousand  Eight  Hundred 
Eighty-five  -fifa  Dollars,  with  interest.  c  H  VIDRARD. 

On  this  note  were  the  following  indorsements:  June  30, 

1874,  $50;  Nov.  8, 1874,  $100;  Feb.  5,  1875,  $125;  April  17, 

1875,  $500;  Dec.  1,  1875,  $500.     How  much  remained  un- 
paid Mar.  1,  1876? 


PROBLEMS  IN  INTEREST. 

PROBLEM  I. 

344.  Principal,   rate,   and   time   given,  to   find   the 
interest. 

This  has  already  been  solved.     (See  page  220.) 

RULE. — Multiply  the  interest  of  $1,  for  the  given  rate  and 
time,  by  the  principal. 

PEOBLEM  II. 

345.  Principal,  rate,  and  interest  given,  to  find  the 
time. 

1.  How  much  is  the  interest  of  $100  for  a  year  at  6^  ? 
For  2  years  ?     For  3  years  ? 


PAETIAL    PAYMENTS.  233 


2.  When  $100,  loaned  at  6^,  brings  an  income  of 

for  how  long  a  time  was  it  loaned  ?     How  long  when  the  in- 
terest was  $18?     $24?     $3?     $4?     $2?     $1.50? 

3.  When  $50,  loaned  at  10  ft,  brings  an  income  of  $10, 
for  how  long  a  time  was  it  loaned? 

KULE. — Divide  the  given  interest  by  the  interest  of  the  prin- 
cipal for  .one  year. 

In  what  time  will 

4.  $250  produce  $30  interest  at  §%  ? 

5.  $600  produce  $24  interest  at  8^  ? 

6.  $115  produce  $13.80  interest  at  6^  ? 

7.  $12.60  produce  $4.15  interest  at  1%  ? 

8.  $35.25  produce  $13.25  interest  at  1%  ? 

9.  $25  produce  $25  interest  at  6ft>  ? 

10.  $150  double  itself  at  8^  ? 

11.  Any  sum  double  itself  at  5^  ?     6^  ?     7^  ? 

12.  Any  sum  triple  itself  at  5^  ?     6^  ?     If0  ? 

PROBLEM  III. 

346.  Principal,  time,  and  interest  given,  to  find  the 
rate. 

1.  What  is  the  interest  of  $100  for  1  year  at  lfc  ?    At 
2^  ?    At  3^  ? 

2.  When  the  interest  of  $100  for  1  year  was  $8,  what  was 
the  rate? 

3.  When  the  interest  of  $100  for  2  years  was  $14,  what 
was  the  rate? 

4.  When  the  interest  of  $50  for  3  years  was  $15,  what' 
was  the  rate? 

RULE. — Divide  the  given  interest,  by  the  interest  of  the  prin- 
cipal for  the  given  time,  at  1  per  cent. 


234  PERCENTAGE. 

What  is  the  rate  per  cent,  when  the  interest 

5.  Of  $125  for  2  years  is  $15? 

6.  Of  $250  for  6  months  is  $8.75? 

7.  Of  $415  for  2  years  6  months  is  $56.025? 

8.  Of  $317  for  1  year  5  months  is  $31.44? 

9.  Of  $215  for  2  years  7  months  10  days  is  $39.30? 

10.  Of  $325.18  for  5  months  26  days  is  $11.13? 

11.  Of  $30.18  for  63  days  is  $  .32? 

12.  Of  $24.36  for  93  days  is  $  .44? 

13.  Of  $25.40  for  45  days  is  $  .397  ? 


PKOBLEM  IV. 

347.  Rate,  time,  and  interest  given,  to  find  the  prin- 
cipal. 

1.  At  6^,  what  sum  will  produce  $6  yearly? 

2.  At  6^g,   what  sum  will   produce  $12  yearly?     What 
$18   yearly?     What  $12   in  two  years?     What  $18  in  2 
years  ? 

RULE. — Divide  the  given  interest  by  the  interest  of  $1,  for  the 
given  time  at  the  given  rate. 

What  sum  of  money  will  produce 

3.  $36.60  interest  in  2  years  at  6^  ? 

4.  $35.70  interest  in  2  years  6  months  at  8^  ? 

5.  $51.20  interest  in  5  years  6  months  at  5  ft  ? 

6.  $50.84  interest  in  6  months  27  days  at  6^  ? 

7.  $39.18  interest  in  5  months  18  days  at  6^  ? 

8.  $41.25  interest  in  3  months  15  days  at  9^  ? 

9.  $87.50  interest  in  1  month    12  days  at  l<fc  ? 

10.  $68.75  interest  in  3  months  10  days  at  6^  ? 

11.  $50.83  interest  in  2  years  3  months  at  6^  ? 

12.  $81.25  interest  in  3  years  5  months  at  8^  ? 


PEKCENTAGE.  235 


NOTES. 

348.  A  Note,  or  Promissory  Note,  is  a  written 
promise  to  pay  a  sum  of  money  at  a  given  time. 

349.  The  Maker  or  Drawer  is  the  person  who  signs 
the  note. 

350.  The  Payee  is  the   person   to  whom   it  is   made 
payable. 

351.  The  Solder  is  the  person  who  has  the  note  in 

"  i  possession. 

352.  The  Indorser  is  the  person  who  writes  his  name 
upon  the  back  of  the  note  to  transfer  it  or  guarantee  its 
payment. 

The  payee  may  indorse  by  writing  his  name  on  the  back  of  a  note. 
It  is  then  payable  to  the  bearer.  He  may  also  indorse  by  writing, 

"Pay  to  A B ,  or  order."     It  then  requires  the  signature  of 

A B before  it  is  payable. 

353.  The  Face  of  a  note  is  the  sum  named  in  it. 

354.  A  Negotiable  Note  is  a  note  payable  to  the 
order  of  the  payee  or  bearer. 

355.  A  Non-negotiable  Note  is  a  note  payable  to 
the  payee  only. 

Notes  should  contain:  the  date,  the  time  when  due,  the  amount  of 
the  note  written  in  words,  the  words  "  for  value  received,"  and  "  with 
interest,"  if  such  is  the  contract. 

356.  Three  Days  of  Grace  are  usually  allowed  after 
the  specified  time,  before  the  note  is  said  to  mature  or  be  due, 
except  on  notes  payable  on  demand,  and  where  the  words 
"  without  grace"  are  written. 


236  PERCENTAGE. 

If,  when  a  note  is  unpaid  at  its  maturity,  the  holder  fails 
to  notify  the  indorsers  of  the  fact,  they  are  released  from 
responsibility  regarding  its  payment. 

357.  Notes  without  interest  draw  interest  at  the  legal  rate 
after  they  become  due,  but  a  note  does  not  draw  interest 
until  after  it  is  due,  unless  the  words  "with  interest,"  or 
' '  with  use,"  are  written  in  it. 

FORMS  OF  NEGOTIABLE  NOTES. 

$729.18.  CINCINNATI,  O.,  Oct.  5,  1876. 

For  value  received,  two  months  after  date  I  promise  to  pay 
James  J.  Cone,  or  order,  Seven  Hundred  Twenty-Nine  -^fo  Dol- 
lars, with  interest.  H  R  BUCKHAM. 

$600.  DETROIT,  Jan.  29,  1877. 

For  value  received,  three  months  after  date  I  promise  to  pay 
H.  O.  Burlingame,  or  bearer,  Six  Hundred  Dollars. 

W.  H.  SARGENT. 

WRITTEN    EXERCISES. 

1.  Write  a  negotiable  note  for  $500.25,  making  yourself 
the  payee,  and  James  J.  Rogers  the  maker.     Interest  at  the 
legal  rate. 

2.  Write  a  non-negotiable  note  for  $315.17,  making  W.  R. 
Howard  the  payee,  payable  on  demand  without  interest. 

3.  Write  two  forms  of  negotiable  notes  for  $3184.25,  due 
in  three  months  to  James  P.  Hermann,  with  interest. 

4.  Indorse  them  properly  for  transferring;   one  to  bearer, 
and  the  other  to  H.  H.  Hurd,  or  order. 

5.  Write  a  note  from  the  following  data:    Face,  $5000; 
negotiable;   maker,  P.  G.  Sloane;   payee,  J.  S.  Orton;   pay- 
able on  demand ;   rate  of  interest,  the  legal  rate. 


DISCOUNT.  237 


COMMERCIAL  DISCOUNT. 

358.  Discount  is   an  allowance,   or  deduction,   made 
from  a  sum  of  money  to  be  paid. 

Problems  in  discount  may,  or  may  not,  have  reference  to 
time. 

359.  Commercial  Discount  is  a  deduction  from 
the  price  of  an  article,  or  from  a  bill,  without  regard  to  time. 

360.  The   Net  Price  is  the  selling  price,  minus  the 
discount. 

361.  The  Cash  Value  of  a  bill,  is  its  face  less  the  dis- 
count. 

EXERCISES. 

362.  1.  If  gloves  marked  at  $1.50  per  pair,  are  sold  at 
10%  discount,  what  is  the  discount?     What  is  the  net  price? 

2.  If  flour  is  offered  at  $7.50  per  barrel,  with  a  discount 
of  5%  for  cash,  what  is  the  discount? 

3.  Find  the  value  of  a  bill  of  goods,  amounting  to  $845 
at  5^  discount. 

4.  Which  is  more  profitable,  to  buy  on  60  days  credit,  or 
at  \\%  off  for  cash,  money  being  worth  1%  ? 

5.  What  is  the  cash  value  of  a  bill  of  goods,  amounting 
to  $3215.45  at  20^  discount,  and  5^5  off  for  cash? 

In  problems  like  No.  5,  it  is  always  understood  that  the  per  cent. 
off  for  cash  is  to  be  reckoned  upon  the  price  after  the  previous  discounts 
have  been  allowed. 

6.  What  is  the  cash  value  of  a  bill  amounting  to  $3750 
at  10%  discount,  and  2$  ft  off  for  cash? 

7.  What  is  the  cash  value  of  a  bill  of  goods  amounting  to 
$2157.25  at  15%  discount,  and  3%  off  for  cash? 


238  PERCENTAGE. 


TRUE  DISCOUNT. 

363.  1.  What  will  be  the  amount  of  $100  in  1  year,  in- 
terest at  6^  ?     In  2  years  ?     In  3  years  ?     In  4  years  ? 

2.  What  is   the  value  now  of  $106,  to  be  paid  in  1  yr., 
when  money  loans  at  6^  ?     Of  $112,  to  be  paid  in  2  yr.  ? 

3.  What  is   the  value  now  of  $212,  to  be  paid  in  1  yr., 
money  loaning  at  6^  ?     Of  $224,  to   be   paid  in  2  yr.  ? 

4.  What  is  the  worth  now  of  a  debt  of  $535,  to  be  paid 
in  1  yr.,  when  money  can  be  loaned  at  7^  ? 

5.  What  is  the  present  value  of  $672,  due  in  1-J-  yr.,  when 
money  is  loaning  at  8^?     Of  $316,  due  in  2  yr.,  money 
being  worth  8fo  ? 

364.  True  Discount  is  a  deduction  made  for  the  pay- 
ment of  a  sum  of  money  before  it  is  due. 

365.  The  Present  Worth  of  a  sum  of  money  due  at 
some   future   time,  is  a  sum  which,  put  at  interest  at  the 
specified  rate,  will  amount  to  the  debt  when  it  becomes  due. 

WRITTEN    EXERCISES. 

366.  1.    What   sum    of   ready    money    is   equivalent    to 
$784.25,  payable  in  2  years,  when  money  is  worth  7^  ? 

PROCESS.  ANALYSIS.  —  Since 

dollar  if  ?" 


$1.14=  Amount  of  $1  for  2  yr. 

interest    now    at    icfc 

8784.25-f-1.14  =  *687.94.  would  amountto$1./4 

$687.  94  =  Present  Worth.  in  2  years,  it  will  re- 

$784.25  —  $687.94=496.31  Discount.        q^e  as  many  dollars 

now,     to     amount     to 

$784.25  in  2  years,  as  $1.14  is  contained  times  in  $784.25,  which  is 
687.94  times.     Therefore,  the  present  worth  is  $687.94. 

The  face  of  the  debt,  $784.25  —  $687.94,  leaves  $96.31,  the  discount. 


DISCOUNT.  239 

RULE.  —  Divide  the  amount  due  by  the  amount  of  $1,  for  the 
given  time  and  rate,  and  the  quotient  will  be  the  present  worth. 

Subtract  the  present  worth  from  the  amount  due,  and  the  re- 
mainder will  be  the  true  discount. 

What  are  the  present  worth  and  discount  of  the  following  : 

2.  $975.50,  payable  in  1|-  years,  when  money  is  worth  6^  ? 

3.  $845.20,  payable  in  1^  years,  when  money  is  worth  1%  ? 

4.  $958.75,  payable  in  3    years,  when  money  is  worth  7%  ? 

5.  $576.25,  payable  in  3    years,  when  money  is  worth  8^  ? 

6.  $8575,  payable  in  2  yr.  3  mo.   10  da.,  when  money  is 
worth  5^  ? 

7.  $4274,  payable  in  1  yr.  4  mo.  15  da.,  when  money  is 
worth  6^  ? 

8.  $2845,  payable  in  3  yr.  6  mo.  15  da.,  when  money  is 
worth  7^  ? 

9.  $1752.75,  payable  in  1  yr.  3  mo.  20  da.,  when  money  is 
worth  8fi  ? 

10.  $5493.50,  payable  in  2  yr.  5  mo.  25  da.,  when  money  is 
worth  9^  ? 

11.  $3457.84,  payable  in  7  mo.  10  da.,  when  money  is 
worth 


12.  Bought  a   farm    for    $14500    cash,   and    sold   it    for 
$16000,   payable  one-half  cash    and    the    remainder  in   one 
year.     To  what  amount  of  cash  in  hand  was  the  selling  price 
equal  when  money  was  loaning  at  6^  ? 

13.  A  merchant  bought  a  bill  of  goods  amounting  to  $5275, 
on  3  mo.  credit,  but  was  offered  2%fo  discount  for  cash.     How 
much  would  he  gain  by  paying  cash,  money  being  worth  8%  ? 

14.  A  merchant   holds  two  notes,   one   for   $187.25,    due 
Feb.   15th,  1877,  and  the  other  for  $382.75,  due  April  1st, 
1877.     What  would   be  due   him   in   cash,   on  both   notes, 
Jan.  1st,  1877,  money  being  worth  8%  ? 


240  PERCENTAGE. 

15.  If  I  pay  a  debt  of  $4725  1  yr.  3  mo.  15  da.  before  it 
is  due,  what  discount  should  be  made  from  the  face  of  the 
debt,  money  being  worth  8^  ? 

16.  Bought  850  barrels  of  pork  at  $21.50  per  barrel,  on 
90  days  credit,  and  sold  it  the  same  day  for  $21.50  per  barrel, 
cash.     How  much  did  I  gain,  money  being  worth  6-^  ? 

17.  How  much  will  I  gain  if,  instead  of  paying  $5400, 
cash,  for  a  piece  of  property,  I  pay  $6000  in  1  yr.  4  mo., 
money  being  worth  9^  ? 

18.  How  much  should  be  discounted  on  a  bill  of  $3725.87, 
due  in  8  mo.  10  da.,  if  it  is  paid  immediately,  money  being 
worth  10^  ? 

19.  Bought  250  barrels  of  granulated  sugar,  weighing  282 
Ib.  each,  at  10f  cents  per  Ib. ,  on  30  days'  credit.     How  much 
cash  would  settle  the  account,  money  being  worth  8  %? 

20.  A  crockery  dealer  bought  $3725  worth  of  goods,  for 
cash,  at  a  discount  of  37^  from  the  regular  list  prices,  and 
sold  them  at  the  regular  list  prices  on  4  months'  time.     How 
much  did  he  gain,  money  being  worth  8%  ? 


BANK  DISCOUNT. 

367.  A  Sank  is  an  institution  established  by  law  to 
receive  money  for  safe-keeping,  to  loan  money,  or  to  issue 
notes  or  bills  to  circulate  as  money. 

368.  A  Check  is  a  written  order  upon  a  bank  for  money 
deposited  with  it. 

369.  Sank  Discount  is  the  simple  interest  on   the 
amount  due,   paid  in  advance,  for   3  days   more  than   the 
specified  time. 

370.  The   Proceeds,  or  Avails,  of  a  note  are  the 
face  of  the  note,  less  the  discount. 


BANK    DISCOUNT.  241 

371.  The   Maturity  of  a  note  occurs  on  the  last  day 
of  grace. 

If  the  last  day  of  grace  falls  on  Sunday  or  a  legal  holiday  the  note 
matures  one  day  earlier. 

372.  The  Term  of  Discount  is  the  number  of  days 
from  the  time  of  discounting  to  the  maturity  of  the  note. 

1.  Banks  usually  discount  for  short  terms,  not  exceeding  3  mo.  or 

4  months. 

2.  Unless  otherwise  specified,  30  days  are  a  month ;  65  days,  2  mo. 

5  da.,  etc. 

3.  Bank  discount  is  undoubtedly  wrong  in  principle,  but  has  been 
sanctioned  by  custom. 

CASE  I. 

373.  To  find  the  Bank  Discount  and  Proceeds* 

1.  What  is  the  bank  discount  of  6385  for  3  mo.  at  6^  ? 

PROCESS.  ANALYSIS.  —  Since 

$385,  Face  of  Note.  banks> in  Discounting, 

A -i  rr:    T>  x       e  TV  require  the  simple  in- 

.0155,  Kate  of  Discount.  .       , 

terest  in  advance,  we 


$5.9675,  Discount.  find   the    interest    of 

$385 — $5. 9675  =  $379. 0325,  Proceeds.     $385  for  3  mo- 3  da-  at 

6^,  which  is  $5.9675. 

And  since  the  present  worth  is  the  face  of  the  debt  minus  the  dis- 
count, we  subtract  $5.9675  from  $385,  which  gives  $379.0325,  the  pro- 
ceeds or  avails. 

KULE. — To  find  the  bank  discount:  Compute  the  interest  on 
the  face  of  the  note  for  3  days  more  than  the  specified  time  at  the 
given  rate. 

To  find  the  proceeds :  Subtract  the  bank  discount  from  the  face 
of  the  note. 

If  the  note  bears  interest,  find  the  discount  on  the  amount  of  the 
note  at  its  maturity.  This  cmiountj  less  the  discount,  will  be  the  pro- 
ceeds. 


242  PERCENTAGE 

Find  the  bank  discount  of: 

2.  $  318.25  for  2  mo.  15  da.  at 

3.  $3846.18  for  1  mo.  17  da.  at 

4.  $2184.39  for  3  mo.  10  da.  at 

5.  $  824.17  for  3  mo.  28  da.  at 

6.  $4484.15  for  3  mo.  18  da.  at 

7.  $7318.69  for  2  mo.  15  da.  at 

8.  $8984.15  for  30  da.  at 

9.  $  765.30  for  65  da.  at 

10.  $375.  CHICAGO,  ILL.,  Dec.  20,  1876. 
Two  months  after  date,  for  value  received  I  promise  to  pay  E. 

D.  Bronson,  or  order,  Three  Hundred  Seventy- Five  Dollars, 
with  interest  at  10^,  at  the  First  National  Bank,  Chicago,  III 

S.  HOWARD  BLACKWELL. 

This  note  was  discounted  Jan.  23,  1877,  at  10%.     What 
was  the  bank  discount?     What  were  the  proceeds? 

11.  What  is  the  bank  discount  on  a  note  for  $890.25  at 
2  months,  the  rate  being  1%  ?    What  are  the  proceeds? 

12.  What  is  the    difference  between    the    bank   discount 
and  the  true  discount,  on  a  note  for  3  months  for  $3725.85, 
the  rate  being  8  %  ? 

13.  $15,725.95.  NEW  YORK,  May  15,  1876. 
For  value  received,  two  months  after  date  "  The  North  River 

Sugar  Refining  Co."  promises  to  pay  Messrs.  Smith  &  Haughton, 
or  order,  Fifteen  Thousand  Seven  Hundred  Twenty-five  T9^- 
Dollars,  at  The  Merchants  National  Bank,  New  Orleans,  La. 

NORTH  RIVER  SUGAR  REFINING  Co., 
BY  C.  MOLLER,  Sec'y. 

The  above  note  was  discounted   May  25,  1876,  at  6%. 
What  was  the  discount?     What  were  the  proceeds? 


BANK    DISCOUNT.  243 

14.  Mr.  A  sold  goods  amounting  to  $3782.75,  and  received 
in  payment  a  note  at  90  days,  without  interest,  which  he  had 
discounted  at  a  bank  after  holding  it  30  days.     How  much 
did  he  realize  from  the- sale,  the  rate  of  discount  being  8%  ? 

15.  A   merchant  sold   32   yards  of  silk  velvet  at   $8.75 
per  yard;   48  yards  of  silk  at  $2.80  per  yard;   25  yards  of 
English  broadcloth  at  $6.25  per  yard;   receiving  in  payment 
a  bank-note   payable  in  45  days,  without  interest,  which  he 
had  discounted  on  the  same  day  at  9%.     How  much  did 
he  realize  in  cash  from  the  sale? 

CASE  II. 

374.  To  find  the  face  of  a  note  when  the  proceeds, 
time,  and  rate  are  given. 

1.  What  is  the  bank  discount  of  $1  for  2  mo.  27  da.  at 
6%?     What  the  proceeds? 

2.  Since  $.985  is  the  proceeds  of  $1  when  discounted  at 
2  mo.  27  da.  at  6%,  of  what  sum  is  $1.97  the  proceeds  for 
the  same  time  and  rate?     Of  what  sum  is  $2.955  the  pro- 
ceeds for  the  same  time  and  rate? 

3.  Of  what  sum  is  $394  the  proceeds,  when  discounted  at 
a  bank  for  1  mo.  27  da.  at  6%?     Of  what  sum  is  $197  the 
proceeds  ? 

4.  Of  what  sum  is  $396  the  proceeds,  when  discounted  at 
a  bank  for  1  mo.  27  da.  at  6%  ? 

5.  The  proceeds  of  a  note  discounted  at  a  bank  for  2  mo. 
12  da.  at  1%  were  $1182.50.     What  was  the  face  of  it? 

PROCESS.  ANALYSIS. — 

$1— $.0145f  =  $.98541  The  proceeds  of  «1.         Since  the  l3™' 

$1182.50--.9854|-=:$1200        The  face  of  the  note.      Ceedl  J** 

are     $  .9854J, 

$1182.50  are  the  proceeds  of  as  many  dollars  as  $  .9854 \  is  contained 
times  in  $1182.50,  which  is  1200  times.  Therefore  the  face  of  the  note 
was  $1200. 


244  PEKCENTAGE. 

RULE. — Divide  the  proceeds  by  the  proceeds  of  $1,  at  the  given 
rate  for  3  days  more  than  the  specified  time. 

6.  The  proceeds  of  a  note  discounted  at  a  bank  for  2  mo. 
at  6%  are  $989.50.     What  was  its  face? 

7.  What  must  be  the  face  of  a  note  so  that  when  dis- 
counted at  a  bank  for  3  mo.  at  6%,  the  proceeds  will  be 
$1969? 

8.  A  merchant  wishes  to  use  $975,  which  he  can  secure 
by  giving  a  bank-note  payable  in  60  days,  to  be  discounted 
at  1%.     For  what  sum  must  the  note  be  written? 

9.  For  what  sum  must  a  note  be  drawn  payable  in  90  days, 
so  that  when  discounted  at  6%  the  proceeds  may  be  $1000? 

10.  A  man  owes  a  debt  of  $1375.38,  which  he  can  meet 
by  giving  his  note  payable  in  3  mo.,  discounted  at  9^.     For 
what  sum  must  the  note  be  written  so  that  the  avails  may 
be  just  large  enough  to  discharge  the  debt? 

11.  For  what  sum  must  a  note  be  drawn,  payable  in  15 
days,  so  that  when  discounted  at  8fo   the  proceeds  may  be 
$1257.25? 

12.  A  speculator  wishes  to  raise  $5250  on  an  indorsed 
note,  payable   in  45  days,   discounted  at  7^.     For  what 
amount  must  he  draw  the  note  so  that  the  proceeds  may  be 
exactly  that  sum? 

13.  For  what  amount  must  a  note  be  made  payable  in  3 
months,  and  discounted  at  ¥l\%  so  that  the  proceeds  may  be 
$1875? 

14.  A  merchant  wishes  to  raise  $500  at  a  bank  by  a  note 
at  60  days.     For  what  sum  must  the  note  be  drawn  that  he 
may  receive  $500  in  cash  from  the  banker  after  paying  the 
discount  at  8%  ? 

15.  For  how  large  a  sum  must  a  note  be  drawn,  payable 
in  90  days,  that  the  net  proceeds  may  be  $15000  after  de- 
ducting the  bank  discount  at  8  fa  ? 


PERCENTAGE.  245 


REVIEW  EXERCISES. 

375.    1.  What  sum   must  be  invested  at  8  ft  to  yield  an 
annual  income  of  $1400? 

2.  A  merchant  bought  a  bill  of  goods  amounting  to  $7825 
on  90  days  credit,  but  was  offered  a  discount  for  cash  of  4^. 
What  was  the  difference  in  the  offers,  money  being  worth 
9^? 

3.  A  man  bought  a  farm  of  135  A.  25  sq.  rd.  for  $62.50 
per  acre.     He  paid  one-third  of  the  purchase  money  in  cash  ; 
one-half  the  remainder  in  6   mo.,  and  the  balance  in  1  yr. 
3  mo.     Money  being  worth  6^,  what  would  have  been  the 
cash  price  of  the  farm? 

4.  A  will  contained  a  bequest  of  $4500  to  a  charity  hos- 
pital, to  be  paid  in  1  yr.  3  mo.     Money  being  worth  7%, 
what  was  the  cash  value  of  the  bequest? 

5.  What  is  the  amount  of  $3752  for  3  yr.  2  mo.  15  da., 
with  compound  interest  at  7  %  ? 

6.  $175.  CINCINNATI,  O.,  March  25,  1872. 
For  value  received,  on  demand  I  promise  to  pay  J.  H. 

Sheppard,    or  bearer,    One   Hundred    Seventy-five   Dollars, 
with  interest  at  8%.  DAvn>  R  AsPELL. 

This  note  was  paid  April  15,  1876.     How  much  was  due 
upon  it? 

7.  What  is  the  difference  between  the  true  discount  and 
the  bank  discount  of  $5728  for  90  days  at  8%  ? 

8.  On  a  note  of  $3729.75,  dated  Feb.   20,  1872,  bearing 
6^  interest,  were  the  following  indorsements;  July  15,  1872, 
$525;  Dec.   15,  1872,  $478;  Feb.   20,  1873,  $25;  May  17, 
1873,  $75;  Sept.  28,  1873,  $1000.     What  was  due  Jan.  15, 
1874? 


246  PERCENTAGE. 

9.  A  coal  dealer  bought  8790  T.  of  coal  at  $3.75  per  T. 
He  sold  15  ft  of  it  at  a  gain  of  10^  on  the  cost,  40%  of  it 
at  a  gain  of  5%,  and  the  rest  at  a  gain  of  8  fa.     He  paid 
3i^  of  the  cost  for  transportation.     How  much  did  he  gain 
by  the  speculation  ? 

10.  A  grain  speculator  bought  10000  bushels  of  barley,  at 
85  cents  per  bushel  cash.     He  sold  it  the  same  day  at  an 
advance  of  4^ ,  receiving  in  payment  a  note  at  30  days  which 
he  had  discounted  at  a  bank  at  9^.     What  was  his  gain  in 
cash? 

11.  A  man  bought  a  house,  paying  62|^  of  the  price  in 
cash,  and  the  rest  in  notes  to  the  amount  of  $3300.     What 
was  the  cost  of  the  house? 

12.  A  merchant  sold  15  ft  of  his  stock  of  dry  goods  the 
first  month,  10^  the  second  month  and  25^  of  the  remainder 
the  third  month,  when  he  took  an  inventory  of  his  stock  on 
hand,  and  found  that  he  had  remaining  $5300  worth  of  goods. 
What  was  the  original  value  of  the  stock  ? 

13.  What  number  is  that  to  which,  if  f  of  25%  of  |  of  480 
be  added,  the  sum  will  equal  25%  of  f  of  50%  of  324? 

14.  A  speculator  bought   1000  bbl.    of  flour  at  a  given 
price  per  bbl.  paying  f  of  its  value  in  cash  and  giving  a  bank- 
note at  60  days  for  the  balance,  which  was  discounted  on  the 
day  it  was  given  at  6% .     The  discount  on  the  note  was  $31. 50. 
How  much  did  he  pay  for  the  flour  per  bbl.  ? 

15.  What  is  the  difference  between  the  simple  and  the  com- 
pound interest  of  $4725.50  for  2  yr.  2  mo.  15  da.  at  6%. 

16.  What  is  the  difference  between  the  amount  of  $3240 
for  5  yr.  3  mo.  10  da.  at  7%  simple  interest,  and  the  amount 
of  the  same  sum  for  the  same  time  and  rate,  with  interest 
payable  annually? 

17.  A  gentleman  invested  -|-  of  his  annual  income  in  mort- 
gages, paying  6^  annual  interest.     In  6  mo.  12  da.  his  in- 
terest from  them  was  $640.    What  was  his  annual  income? 


PROFIT   AND    LOSS.  247 


PROFIT  AND  LOSS. 

376.    1.  When  25^  is  gained  what  part  is  gained? 

2.  I  sold  a  coat  which  cost  me  $12  for  25^  more  than  it 
cost.     How  much  did  I  gain  ?     How  much  did  I  get  for  it? 

3.  Paid  $15  for  a  ton  of  hay,  and  sold  it  at  a  loss  of  20%. 
How  much  did  I  lose  2     How  much  did  I  get  for  the  hay  ? 

4.  If  I  sell  land  that  cost  me  $50  an  acre,  at  an  advance  of 
^ ,  how  much  do  I  get  per  acre  ? 

5.  If  I  paid  $50  per  acre  for  my  land  and  sell  it  at  $5  per 
acre  less  than  I  paid,  what  part  of  the  cost  do  I  lose  ?    What  %  ? 

6.  If  I  sell  flour  for  $8  per  bbl.  that  cost  me  $6,  what  part 
of  the  cost  do  I  gain  ?     What  per  cent.  ? 

7.  Sold  boots  at  $5  a  pair  that  cost  $4,  what  part  of  the  cost 
was  gained  ?     What  per  cent.  ? 

8.  By  selling  flour  at  a  profit  of  $2  per  bbl.,  20^,  or  |,  of 
the  cost  was  gained.     What  was  the  cost  ? 

9.  Sold  wheat  at  a  profit  of  $  .10  per  bu.  which  was  5^  of 
its  cost.     What  was  its  cost? 

10.  If  I  sell  a  cow  that  cost  me  $50,  at  an  advance  of  20^ 
on  the  cost,  how  much  will  my  profit  be  ?     How  much  do  I 
get  for  her? 

11.  By  selling  flour  at  $10  per  bbl.,  a  profit  of  25^  was 
made.     What  did  it  cost  ? 

ANALYSIS. — Since  25^  or  \  of  the  cost  was  gained,  the  selling  price 
must  have  been  f  of  the  cost;  and  since  f  of  the  cost  was  $10,  J  of  the 
cost  is  \  of  $10,  which  is  $2 ;  and  since  \  of  the  cost  is  $2,  the  cost  is  4 
times  $2  or  $8. 

12.  By  selling  dress  goods  at  66  cents  per  yd.,  a  profit  of 
was  made.     What  was  the  cost? 

13.  By  selling  tea  at  $  .80  a  pound,  a  loss  of  20^  was  in- 
curred.    What  was  the  cost  ? 


248  PERCENTAGE. 

14.  If  I  get  $60  for  a  cow,  and  thereby  gain  20^,  or  |  of 
the  cost,  what  did  she  cost  me? 

15.  I  bought  a  horse  for  $150  and  sold  him  at  an  advance 
of  20^ .     What  did  I  get  for  him  ? 

DEFINITIONS. 

377.  Profit  and  Loss  are  terms  used  to  denote  the  gain 
or  loss  in  business. 

378.  The  processes  in  Profit  and  Loss  involve  the  same 
elements  as  do  the  fundamental  problems  in  Percentage.    The 
corresponding  terms  are: 

1.  The  Cost,  or  Sam  Invested,  is  the  Base. 

2.  The  Rate  Per  Cent,  of  profit  or  loss  is  the  Kate. 

3.  The  Gain  or  Loss  is  the  Percentage. 

4.  The   Selling  Price,  when  more   than  the  cost,   is  the 
Amount. 

5.  The   Selling  Price,   when   less    than    the    cost,    is    the 
Difference. 

PRINCIPLE. — The  gain  or  loss  is  reckoned  at  a  certain  rate  per 
cent,  on  the  cost  or  sum  invested. 


WRITTEN    EXERCISES. 

379.    1.  A  paid  $3500  for  a  house,  and  sold  it  at  10^  ad- 
vance.   How  much  did  he  gain?    How  much  did  he  get  for  it? 
PROCESS.  ANALYSIS. — Since  the 

$3500  X     .10  =  $  350,  Gain.  honse  ™s  sold  atuan  ad' 

vance  of  10 <fi>  °n  tne  cost> 
$3500  +  $350  =f=  $3850,  Selling  price.     his  gain  was  _i_o_  or  A 

Cost  X  Rate  =  Gain.  of  $3500,  which  is  $350; 

and  the  selling  price   is 
equal  to  the  sum  of  the  cost  and  gain,  or  $3850. 


PROFIT    AND    LOSS.  249 

RULES. — Since  the  same  elements  are  involved  as  in  thejun- 
damental  problems  in  Percentage  the  rules  are  the  same. 

FORMULAS. 

1.  Gain  or  loss  —  Cost  X  Rate. 

2.  Rate  =  Gain  or  loss  -+-  Cost. 

3.  Cost  —  Gain  or  loss  -r-  Rate. 

4.  Cost  —  Selling  price  -f-  ( 1  +  Rate  ). 

5.  Cost  —  Selling  price  -f-  ( 1  —  Rate  ). 

2.  Mr.  A.  bought  a  piano  for  $275,  and  sold  it  at  an  ad- 
vance of  25^.     How  much  did  he  receive  for  it? 

3.  A  bookseller  bought  $3584  worth  of  books,   and  sold 
them  at  a  gain  of  lO^J.     How  much  did  he  gain? 

4.  A  carriage  maker  sold  a  carriage  at  25^  advance  on  the 
cost.     It  cost  him  $318.25.     How  much  did  he  get  for  it? 

5.  A  harness  maker  sold  a  set  of  double  harness  at  a  profit 
of  15^.     It  cost  him  $45.     What  did  he  get  for  it? 

6.  A  manufacturer  of  tools  sold  5  dozen  axes  at  a  profit  of 
12^.     They  cost  him  $9  per  dozen.     What  was  his  profit? 

7.  A  merchant  sold  a  bill  of  goods  at  a  profit  of  15^. 
The  goods  cost  him  $84.25.     What  did  he  receive  for  them? 

8.  A  speculator  bought  50000  pounds  of  sole  leather,  which 
he  sold  at  a  profit  of  8^.     If  it  cost  him  $6000,  what  did  he 
get  for  it? 

9.  A  man  sold  his  house  at  a  profit  of  15^.     If  he  paid 
$3000  for  it,  how  much  did  he  get  for  it? 

10.  A  drover  sold  a  flock  of  sheep  at  a  profit  of  7^. 
If  they  cost  him  $1500,  what  did  he  get  for  them? 

11.  A  poultry  dealer  bought  a  quantity  of  poultry,  which 
he  sold  at  a  gain  of  9^.     He  paid  $250  for  it.     How  much 
did  he  get  for  it? 

12.  Mr.  A  bought  cloth  at  $2.15  per  yard.     At  what  price 
must  he  sell  it  to  gain 


250  PERCENTAGE. 

13.  A  farm  which  cost  $65  per  acre  was  sold  at  a  gain  of 

For  how  much  did  it  sell  per  acre? 

14.  A  merchant  bought  3950  yards  of  cotton  at  9^-  cents 
a  yard.     How  much  will  he  get  for  it  if  he  sells  it  at  a  gain 
of  12^! 

15.  A  merchant  desires  to  mark  goods  that  cost  him  $3.60 
per  yard  so  that  he  may  gain  33^^.     At  what  price  must 
the  goods  be  marked? 

16.  A  bankrupt  stock  was  sold  at  35^  loss.     What  was 
the   selling  price   of  articles   that   cost   50c.  ?     $1?     $1.50? 
$1.75? 

17.  What  per  cent,  is  lost  by  selling  sugar  at  10  cents  per 
pound  which  cost  12  cents  per  pound? 

PROCESS.  ANALYSIS. —  Since  sugar  that  cost  12 

<»  -^2 $  10  =  $  02  cents  was  sold  for  10  cents,  there  was  a 

loss  of  2  cents  per  pound.  And  since  the 

*  "^  ~^~  $  '1^  =  1*>3/^  gain  or  loss  is  reckoned  at  a  rate  per  cent. 
Loss  -f-  Cost  =  Rate.  on  the  cost,  we  must  find  what  per  cent.  2 

is  of  12.  2  is  J  of  12;  or,  expressed  as 
hundredths,  is  .16f  of  12,  or  16|y0. 

18.  What  per  cent,   is  gained  by  selling  tea  at  $1  which 
cost  $.75? 

19.  What  per  cent,   is  lost  by  selling  tea  at  $.75  which 
cost  $1? 

20.  What  per  cent,  is  lost  by  selling  cloth  at  $1.25  that 
cost  $1.75? 

21.  Bought  goods  at  50  cents  a  yard  and  sold  them  at  60 
cents  a  yard.     What  per  cent,  did  I  gain? 

22.  Goods  that  are  selling  at  12^-  cents  a  yard  cost  10  cents. 
What  per  cent,  is  gained  by  selling  them  at  that  rate? 

23.  A  man  bought  a  city  lot  for  $4500  and  sold  it  for 
$5000.     What  per  cent,  did  he  gain  ? 

24.  Sold  a  quantity  of  potatoes  for  $850  which  cost  me 
$970.     What  per  cent,  did  I  lose? 


PROFIT    AND    LOSS. 


251 


25.  Bought  a  quantity  of  crude  petroleum  at  5  cents  per 
lion  and  sold  it  at  4-|-  cents.    What  per  cent,  did  I  lose  ? 

26.  A  fruiterer  bought  10  boxes  oranges  at  $1.75  per  box. 
Two  of  the  boxes  were  worthless,  but  he  sold  the  balance  at 
nch  price  that  he  gained  5^  on  the  whole  purchase.     How 
luch  did  he  sell  them  for  per  box?     How  much  did  he  gain 

on  the  purchase? 

27.  I  bought  books  at  10^  discount  from  the  retail  price, 
which  was  $1.50  per  volume,  and  sold  them  at  the  retail 
price.     What  was  my  gain  per  cent.  ? 

28.  An  agent  gets  a  discount  of  40^  from  the  retail  price 
of  articles  and  sells  them  at  the  retail  price.     What  is  his 
gain  per  cent.  ? 

29.  A  merchant  bought  cloth  at  $3.25  per  yard,  and  after 
keeping  it  6  months  sold  it  at  $3.75  per  yard.     What  was 
his  gain  per  cent.,  reckoning  6^  per  annum  for  the  use  of 
money  ? 

30.  Which  is  more  profitable,  and  how  much  per  cent.,  to 
sell  goods  for  cash,  at  once,  at  25^  advance,  or  in  1  year  at 

advance,  money  being  worth  1%  ? 

31.  Mr.  A.  gets  a  discount  of  30^  from  the  retail  or  list 
price  of  goods.     Mr.  B.  gets  a  discount  of  30^  also,  and  5^ 
off  for  cash.     If  both  sell  goods  at  the  list  price,  what  is  each 
one's  gain  per  cent.  ? 

32.  A  merchant  bought  tea  at  20^  less  than  its  market 
value,  and  received  a  discount  of  5^  for  cash.     He  sold  it 
at  an  advance  of  15^  above  its  market  value.     What  was 
his  gain  per  cent.  ? 

33.  By  selling  cloth  at  a  gain  of  12  cents  a  yard,  I  real- 
ized a  gain  of  Sf0  on  the  cost.     What  was  the  cost? 


PROCESS. 

$.12 -T- .08  =  $1.50. 
Gain  -f-  Rate  =  Cost 


ANALYSIS. — Since  the  gain,  12  cents,  is 
g^  or  y^  of  the  cost,  J  of  12  cents,  or 
1J  cents,  is  T^  of  the  cost,  and  the  cost  is 
100  times  1 J  cents,  or  $1.50. 


252  PERCENTAGE. 

34.  I  make  10^   by  selling  tea  at  a  profit  of  10  cents  a 
pound.     What  did  it  cost  me?     What  do  I  sell  it  for? 

35.  Flour  was  sold  at  a  profit  of  $1.50  per  bbl.,  which  was 
16|^  of  the  cost.     What  was  the  cost? 

36.  A  merchant  made  12^  by  selling  cloth  at  an  advance 
of  12  cents  a  yard.     What  did  it  cost? 

37.  By  selling  butter  at  8  cents  a  pound  more  than  cost,  a 
grocer  made  20^.     What  did  he  pay  for  it? 

38.  A  merchant  sold  cloth  which  was  damaged  by  fire,  at 
a  sacrifice  of  22  cents  per  yard,  which  was  40^  of  the  cost. 
What  did  the  goods  cost? 

39.  A  farmer  sold  a  yoke  of  cattle,  to  which  he  had  fed 
$10  worth  of  grain,  at  an  advance  of  $25,  and  still  realized 
a  profit  of  15^.     What  did  they  cost? 

40.  A  man  sold  a  horse  at  an  advance  of  $75,  which  was 
a  gain  of  25^.     What  was  the  cost  of  the  horse? 

41.  If  I  sell  a  quantity  of  apples  at  an  advance  of  25  cents 
a  barrel,  and  thereby  realize  12^  profit,  what  was  the  cost? 

42.  By  selling  cloth  at  a  gain  of  23  cents  per  yard,  I  realize 
a  profit  of  20^.     What  did  it  cost? 

43.  A  merchant  asked  25%  more  for  his  goods  than  they 
cost  him,  but  at  last  sold  them  at  a  reduction  of  10^  from 
his  asking  price,  thus  realizing  from  the  sale  $4684  profit. 
What  was  the  cost  of  the  goods  ? 

.  44.  A  gentleman  sold  a  horse  for  $180  and  gained  20  fo  on 
him.     What  did  the  horse  cost? 

PROCESS.  ANALYSIS. — Since  20^  of 

100% +20%  =120%  !he  co?t  was  gained> the  sel1- 

ing  price  must  have  been  20^, 

$180-5-1.20  =  $150  more  than  the  cost,  or  120^ 

Selling  Price  -f-  (1  +Rate)  =  Cost.    of  the  cost-    And> since  120^ 

of  the  cost  is  $180,  lfc  of  the 

cost  is  TJo  of  $180,  or,  $1.50,  and  the  whole  cost  is  100  times  $1.50, 
or  $150. 

Therefore  the  horse  cost  the  gentleman  $150. 


PKOFIT   AND    LOSS.  253 

45.  A  gentleman  sold  a  carriage  for  $230,  and  thereby 
st  8%  of  the  cost.     What  was  the  cost? 

PROCESS.  ANALYSIS. — Since  8^  of  the 

lOQ^ 8^—92^  °°St  Was  ^°St?  ^e  se^n&  Pr^ce 

*°  must  have  been  8^  less  than 

$230-r-.92  =  $250  t^e  costj  or  92^  Of  the  cost. 

Selling  Price  -r-(l— Rate)  =  Cost.     And>  since  92/*  of  the  cost 

was  $230,  lfc  of  the  cost  was 
jx  of  $230,  or  $2.50,  and  the  whole  cost  was  100  times  $2.50,  or  $250. 

46.  By  selling  apples  at  $.50  per  bushel  a  grocer  gained 
25%  on  the  cost.     What  was  the  cost? 

47.  A  farm  was  sold  for  $38000,  which  was  a  loss  of  5  ft 
of  the  cost.     What  was  the  cost? 

48.  A  block  of  stores  was  sold  for  $185000,  which  was  a 
gain  of  15^.     What  did  they  cost? 

49.  A  merchant  lost  5  ft  by  selling  calico  at  9|-  cents  a 
yard.     What  did  it  cost? 

50.  A  bankrupt  stock  was  sold  for  $3582,  which  was  a  loss 
of  331^.     What  did  it  cost? 

51.  By   selling   molasses   at   65   cents   a   gallon,  a  grocer 
gained  30^.     What  was  its  cost? 

52.  A  man  was  compelled  to  sell  his  household  furniture 
for  $1250,  which  was  a  loss  of  37^.     What  did  it  cost? 

53.  A  boot  and  shoe  dealer  lost  9^  by  selling  boots  at 
$3.75  a  pair.     What  was  the  cost  of  the  boots? 

54.  A  stationer  lost  26^   by  selling  paper  at   $2.22   a 
ream.     What  did  he  pay  for  it? 

t55.  A  druggist  gained  125^  by  selling  alcohol  for  $3.50 
r  gallon.     What  did  he  pay  for  it? 
56.  Coal  was  sold  at  $4.56^-  per  ton,  which  was  a  loss  of 
<fc .     What  was  the  cost  ? 
57.  When  pork  is  selling  at  $4.50  per  hundred-weight  I 
lose  10%.     What  will  be  my  gain  per  cent,  if  I  sell  at  $6 
per  hundred- weight? 


254  PERCENTAGE. 


COMMISSION. 

380.  1.  If  a  man  sells  $500  worth  of  goods  for  me,  how 
much  will  he  receive  if  he  gets  2^  of  the  sales  ? 

2.  If  I  allow  a  man  2^   for  purchasing  $3000  worth  of 
silks  for  me,  how  much  will  he  get  for  his  services  ? 

3.  If  I   collect  a  debt   of  $350,  and  charge  4^    of  the 
amount  for  my  services,  how  much  will  I  receive  ? 

4.  Mr.  B.  paid  his  agent  5^  for  selling  $3000  worth  of 
cotton.     How  much  did  he  pay  him,  or  what  was  his  com- 
mission 1 

5.  At  3^5  commission,  how  much  will  A  receive  for  sell- 
ing $4200  worth  of  flour?     How  much  will  be  left  after  pay- 
ing the  commission,  or  what  will  be  the  net  proceeds? 

6.  If  I  pay  2fo  commission  for  buying  goods,  what  is  the 
cost  of  every  dollar's  worth  of  goods  bought?     Since  every 
dollar's  worth  of  goods  bought  costs  the  purchaser  $1.02,  how 
many  dollars'  worth  can  be  bought  for  $102?     For  $204? 

7.  If  I  pay  3^  commission  for  buying  goods,  how  many 
dollars'  worth  of  goods  can  be  bought  for  $309,  after  paying 
the  commission?     For  $515? 

8.  How  many  dollars'  worth   of  goods  can  be  purchased 
for  $630,  after  making  allowance  for  the  agent's  commission 
at  5^  of  the  value  of  the  goods  purchased? 

DEFINITIONS. 

381.  A   Commission  Merchant  or  Agent  is  a 

person  who  buys  or  sells  goods,  or  transacts  other  business  for 
another. 

382.  The  Commission  is  the  compensation  or  percent- 
age allowed  a  commission  merchant  or  agent. 


COMMISSION.  255 


383.  A  Consignment  is  a  quantity  of  merchandise 
sent  to  a  commission  merchant  or  agent  to  be  sold. 

384:.  The  Consignor  is  the  person  who  sends  the  mer- 
chandise to  be  sold. 

385.  The  Consignee  is  the  person  to  whom  the  mer- 
chandise is  sent. 

386.  The  JVet  Proceeds  of  a  sale  is  the  sum  left  after 
the  commission,  expenses,  etc.,  have  been  deducted. 

387.  The  processes  in  Commission  involve  the  same  ele- 
ments as  do  the  fundamental  problems  in  Percentage.     The 
corresponding  terms  are: 

1.  The  Sales  or  Sum  Invested  is  the  Base. 

2.  The  Rate  Per  Cent,  is  the  Kate. 

3.  The  Commission  is  the  Percentage. 

4.  The  Purehase  Price  plus  the  Commission  is  the  Amount. 

5.  The  Net  Proceeds  is  the  Difference. 

388.  PRINCIPLE. — The  commission  is  reckoned  at  a  certain 
rate  per  cent,  on  the  value  of  the  sales  and  purchases. 

WRITTEN     EXERCISES. 


389.    1.  What  will  be  an  agent's  commission  for  selling 
85.15  worth  of  goods  at  3%fo  ? 

PROCESS.  ANALYSIS. — Since  the  rate 

p-    -i  K  v/    0  3  i $13    48          °^  commissi°n  ig  ^\0/ct  or  .03  J, 

the  commission  will  be  .03  \ 
Sales  X  Rate  =  Commission.          of  $385.15,  which  is  $13.48. 

2.  What  is  the  commission  for-  selling  cattle  to  the  value 

$3184  at  2f  %  ? 
*/v 

3.  What  is  the  commission  for  selling  cotton  to  the  value 

3,  at  21  per  cent.  ? 


256  .PERCENTAGE. 

4.  Mr.  B.  sent  his  agent  $3468  to  invest  in  goods,  allowing 
him  2^  commission.    What  sum  did  he  invest  in  goods  after 
deducting  his  commission  ?    What  was  the  agent's  commission? 

PROCESS.  ANALYSIS. — 

$3468  --1.02  =  $3400,  Amount  invested.     Since   the  5fnt 

gets  a  commission 

Remittance  -=-  (1  -j~  -Rtofe)  =  Purchase  Price.  Of  2%  for  purchas- 
$3468  —  $3400  =  $68,  Commission.  ing,  it  requires 

$1.02   to   purchase 

$1  worth  of  goods ;  he  can  therefore  purchase  as  many  dollars'  worth 
of  goods  for  $3468  as  $1.02  is  contained  times  in  $3468,  which  is  3400 
times.  Therefore,  he  can  purchase  $3400  worth  of  goods.  The  money 
sent  minus  the  amount  invested  will  be  the  commission. 

RULES. — Since  the  same  elements  are  involved  as  in  the  fun- 
damental problems  in  Percentage,  the  rules  are  the  same. 

FORMULAS. 

1.  Commission  =  Sales  or  Purchase  X  Rate. 

2.  Rate  —  Commission  — -  Sales  or  Purchase. 

3.  Sales  or  Purchase  =  Commission  -^-  Rate. 

4.  Purchase  —  Sum  remitted  ~  (  1  -f-  Rate  ). 

5.  Sales  =  Net  Proceeds  -f-  ( 1  —  Rate  ). 

5.  If  I  send  my  agent  $4050  to  invest  in  goods,  after  de- 
ducting 3^  commission,  what  sum  will  he  invest? 

6.  If  I  send  my  agent  $875  to  invest  in  calico,  allowing 
him  2^  commission,  how  many  yards  can  he  buy  at  6  cents 
per  yard? 

7.  How  much  is  an  agent's  commission  for  selling  385  bbl. 
flour  at  $6.50  per  bbl.,  the  rate  of  commission  being  2^  ? 

8.  What   is    the    commission   for    collecting   bills    to    the 
amount  of  $784.25  at  5^  ? 

9.  How  much  must  an  agent  be  paid  for  selling  25  bbl. 
pears  at  $12.75  per  bbl.,  his  commission  being  6^  ? 


COMMISSION.  257 

10.  What  is  the  commission  at   3^   for  selling  125  bbl. 
of  potatoes  at  $2.37|  per  bbl. 

11.  A  commission  merchant  sold  20  firkins  of  butter,  each 
containing  56  Ibs.,  at  28  cents  per  Ib.     How  much  was  his 
commission  at  8^  ? 

12.  What  is  the  commission  at  7^  on  a  sale  of  20  boxes 
of  eggs,  each  containing  22  doz. ,  at  23  cents  per  doz.  ? 

13.  How  much  commission  must  be  paid  to  a  collector  for 
collecting  an  account  of  $928.75  at  3f  ^  ? 

14.  I  sent  my  agent  $1525  to  be  invested  in  goods  after 
deducting  his  commission  of  2^.     How  much  did  he  invest? 

15.  A  merchant  sent  his  agent  $375.50  to  invest  in  muslin 
at  8  cents  per  yard.     After  deducting  3^  commission,  how 
many  yards  of  muslin  did  he  purchase? 

16.  Mr.  A.  sent  $3320.10  to  be  invested  in  goods  after  pay- 
ing his  agent  2^  commission.     What  sum  was  invested? 

17.  A  man  sent  his  agent  $3725.05  to  invest  in  pork  after 
deducting  \\°/c  commission.     How  much  did  he  invest? 

18.  A  speculator  sent  his  agent  in  Chicago  $8966.75  to  in- 
vest in  wheat.     After  deducting  \^0  commission,  how  many 
bushels  of  wheat  did  he  buy  at  $1.11^  per  bushel?     What 
was  the  commission? 

19.  An  agent  received  $24.52  for  selling  goods  at  a  com- 
mission of  2%.     How  many  dollars' worth  of  goods  did  he 
sell? 

20.  How  much  money  must  I  send  my  agent,  so  that  he 
may  purchase  for  me  150  bbl.  flour  at  $8.25  per  bbl.  if  I  pay 
him  3^  commission  for  his  services? 

21.  A  commission  merchant  received  $318.25  for  selling 
$12730  worth  of  bankrupt  goods.     What  was  the  rate  of 
commission  ? 

22.  A  sale  of  real  estate  returned  as  net  proceeds  $2396.49, 
after  paying  $324.18  charges  and  a  commission  of  2^.     For 
"how  much  did  it  sell? 


258  PERCENTAGE. 


REVIEW  EXERCISES. 

390.    1.  A  man  whose  wages  had  been  $27  per  week  was 
obliged  to  take  33^^  less.     How  much  was  the  reduction? 

2.  A  man  bought  a  horse  for  $300  and  sold  him  for  $375. 
What  per  cent,  did  he  gain? 

3.  When  sugars  that  cost  10  cents  a  pound  are  sold  for  11 
cents,  what  per  cent,  is  gained? 

4.  When  land  is  selling  at  an  advance  of  $40  an  acre,  what 
is  the  gain  per  cent,  if  it  cost  $120  an  acre? 

5.  A  boy  sold  apples  at  the  rate  of  2  for  3  cents  which  he 
bought  at  the  rate  of  3  for  2  cents.     What  did  he  gain  per 
cent.  ? 

6.  A  news  agent  sold  $31  worth  of  goods  in  a  day  at  a  com- 
mission of  10^ .     How  much  was  his  commission  ? 

7.  A  man  lost  $80  which  was  just  20^  of  his  money.     How 
much  money  had  he? 

8.  A  man  paid  2^  for  selling  his  wheat  and  realized  $1.47 
per  bushel.     For  how  much  did  it  sell  per  bushel? 

9.  The  interest  on  $240  for  2  years  was  $28.80.     At  what 
per  cent. was  it  loaned? 

10.  When  money  is  loaned  at  6^  and  the  interest  amounts 
to  $75  on  $1000,  how  long  has  it  been  loaned  ? 

11.  When  a  merchant  buys  goods  at  f  of  their  estimated 
value,  and  sells  them  at  their  estimated  value,  how  much  is 
his  gain  per  cent.  ? 

12.  When  a  man  sells  goods  at  a  price  from  which  he  re- 
ceived a  discount  of  30^ ,  what  is  his  gain  per  cent.  ? 

13.  When  a  man  can  borrow  money  at  8^,  which  is  more 
profitable,  and  how  much  per  cent.,  to  buy  goods  at  3^  off 
for  cash  or  at  90  days'  credit  ? 

14.  If  I  sell  a  horse  for  $125  which  cost  me  $175,  what 
do  I  lose  per  cent.  ? 


EEVIEW    EXERCISES.  259 

15.  A  man  sold  a  cow  at  an  advance  of  $10,  which  was 
25%  of  what  she  cost.     How  much  did  she  cost? 

16.  A  boy  can  pick  33-|^  as  many  apples  as  his  father. 
If  his  father  can  pick  18  barrels  a  day,  how  many  can  the 
boy  pick? 

17.  When  a  merchant  buys  goods  at  a  discount  of  20^ 
from  the  regular  price,  and  sells  them  at  20^  more  than  the 
regular  price,  what  is  his  gain  per  cent.  ? 

18.  Which  is  more  profitable,  to  sell  goods  now,  that  cost 
18  cents  a  yard,  for  20  cents  a  yard,  or  to  keep  them  1  year 
and  sell  them  at  21  cents  when  money  is  worth  6^  ?     How 
much  more  profitable  on  an  investment  of  $1000? 

19.  Mr.  A.  bought  a  horse  and  carriage,  paying  twice  as 
much  for  the  horse  as  the  carriage.     He  afterward  sold  the 
horse  for  25^  more  than  he  gave  for  it,  and  the  carriage 
for  20^  less  than  he  gave  for  it,  receiving  for  both  $577.50. 
What  was  the  cost  of  each  ? 

20.  After  getting  a  note,  without  interest,  discounted  at  a 
bank  for  3  months  at  7^,  I  had  $468.39.     What  was  the 
face  of  the  note? 

21.  A  man  can  borrow  money  at  6^   and  pay  cash  for 
goods,  obtaining  a  discount  of  2^,  or  he  may  pay  for  the 
goods  in  60  days.     Which  is  the  more  advantageous,   and 
how  much,  on  an  invoice  of  goods  amounting  to  $1500  ? 

22.  I  had  a  note  for  $1000  discounted  at  a  bank  for  3 
months  at  7^.     The   proceeds  were  invested  in  wheat  at 
$1.65  per  bu.     How  many  bushels  did  I  buy? 

23.  Mr.  A.  sold  a  horse  for  $198,  which  was  10^  less  than 
he  asked  for  him,  and  his  asking  price  was  10^  more  than 
the  horse  cost  him.     What  was  the  cost  of  the  horse  ? 

24.  I  bought  a  horse  of  Mr.  A.  for  20^  less  than  he  cost 
him,  and  I  immediately  sold  the  horse  for  25^  more  than  I 
paid  for  him,  gaining  $25.    What  did  the  horse  cost  Mr.  A.  ? 
What  did  he  cost  me? 


260  PERCENTAGE. 

25.  I  directed  my  agent  to  purchase  for  me  30  village  lots 
at  $650  each,  and  to  pay  the  expenses  of  examining  the  titles, 
which  averaged  $4.25  per  lot.     What  did  they  all  cost  me  if 
the  agent's  commission  was  4^  on  the  price  of  the  lots? 

26.  For  how  much   must  the  lots  be  sold  to  give  me  a 
net  profit  of  20^,  besides  allowing  the  agent  5^  commission 
for  selling? 

27.  After  a  certain  time  a  sum  of  money  which  had  been 
at  interest,  had  increased  18|^  and  amounted  to  $3896.74. 
What  was  the  sum  at  interest? 

28.  A  clothier  sold  two  suits  of  clothes  at  $72  each.     On 
one  he  gained  20^,  and  on  the  other  he  lost  20^.     Did  he 
gain  or  lose  on  the  sale  and  how  much?     How  much  per 
cent.  ? 

29.  After  marking  goods  at  an  advance  of  25^  over  cost, 
a  merchant  made  an  abatement  of  20  ^J   from  the  marked 
price.     Did  he  gain  or  lose,  and  how  much  per  cent.  ? 

30.  On  f  of  my  property  I  gained  33-J^  and  sold  the  rest 
for  -|  of  the  cost  of  the  whole,  receiving  in  payment  a  note  due 
in  3  months  which  I  got  discounted  at  a  bank  at  6^.     What 
was  my  gain  per  cent,  if  my  property  was  worth  $10000? 

31.  I  sent  my  agent  $7000  to  be  invested  in  wheat  at  $1.15 
per  bushel,  allowing  him  a  commission  of  3^  on  the  purchase. 
I  paid  for  storage  2  cents  per  bushel  per  month,  and  \^0  per 
month  for  the  use  of  the  money.     After  3  months  my  agent 
sold  the  wheat  at  $1.33  per  bushel,  charging  2^  commission 
for  making  the  sale,  and  took  in  payment  a  note  for  the  \ 
amount  at  30  days,  which  I  had  discounted  at  a  bank  at  9^. 
Did  I  make  or  lose,  and  how  much? 

32.  A  man  wishing  to  sell  a  horse  and  'cow,  asked  three 
times  as  much  for  the  horse  as  the  cow,  but,  finding  no  pur- 
chaser, he  reduced  the  price  of  the  horse  20^,  and  the  price 
of  the  cow  10%,  and  sold  them  both  for  $165.     What  did 
he  get  for  each? 


REVIEW    EXERCISES.  261 

33.  I  owe  B  66-f  %  of  the  amount  I  owe  A,  and  I  owe  C 
40%  of  the  amount  I  owe  B.     How  much  do  I  owe  each  if 
I  owe  B  $80  more  than  I  do  C  ? 

34.  I  bought  a  quantity  of  coffee  at  28-J-  cents  per  pound. 
Allowing  the  coffee  to  fall  short  5^  in  weighing,  and  10^  of 
the  sales  to  be  lost  through  bad  debts,  for  how  much  must  I 
sell  it  per  pound  that  I  may  make  a  clear  gain  of  20%  on 
the  cost? 

35.  A  quantity  of  prints  was  sold  at  a  commission  of  2^ 
and  the   proceeds  invested   in  cambrics   purchased  at  3% 
commission.     The  commission  for  buying  the  cambrics  was 
$126.30.     How  much  did  the  prints  sell  for? 

36.  A  tailor  sold  a  suit  of  clothes  for  $46,  thereby  gaining 
15^.     He  sold  another  for  $60,  and  lost  the  same  amount 
of  money  which  he  gained  upon  the  first  suit.     What  per 
cent,  did  he  lose  upon  the  last  suit  sold? 

37.  A  merchant  having  a  quantity  of  pork  asked  33^^ 
more  than  it  cost  him,  but  was  obliged  to  sell  it  12|-  per 
cent,  less  than  his  asking  price.     If  he  received  $7  per  cwt., 
what  was  its  cost? 

38.  What  must  be  asked  for  apples  which  cost  $3  per  bbl., 
that  I  may  reduce  my  asking  price  20^ ,  and  still  gain  20^ 
on  the  cost? 

39.  A  merchant  sold  a  consignment  of  blankets   at   3^ 
commission,  receiving  in  payment  a  bank-note  at  30  days, 
which    he   had  discounted  at   6^.     He    then    invested  the 
proceeds  in  wool  at  30  cents  per  lb.,  charging  \<fa   commis- 
sion for  buying  it.     His  commission  for  purchasing  the  wool 
was  $45.     What  was  the  value  of  the  consignment?     How 
much  wool  did  he  buy? 

40.  A  merchant  sold  a  quantity  of  goods  at  a  gain  of  20^ . 
If,  however,  he  had  purchased  the  goods  for  $60  less  than  he 
did,  his  gain  would  have  been  25^.     What  did  the  goods 
cost? 


262  PERCENTAGE. 


TAXES. 

CASE  I. 

GENERAL   TAXES. 

391.  1.  If  a  person  has  to  pay  annually,  for  public  pur- 
poses, 2^  of  the  value  of  his  property,  estimated  at  $5000, 
how  much  will  be  his  tax  1 

2.  If  I  am  taxed  1-g-^  on  my  land,  houses,  etc.,  or  real 
estate,  estimated  at  $20000,  how  much  will  be  my  tax? 

3.  If  I  am  taxed  1%   on  the  value  of  my  movable  or 
personal  property,  what  is  the  tax  on  $6000? 

4.  My  property  is  estimated  by  the  assessors  to  be  worth 
$40000.     What  will  be  my  tax  at  \\%  ? 

5.  I  imported  500  yards  of  silk,  invoiced  at  $2  per  yard. 
What  will  be  the  Government  tax,  or  duty,  upon  it  at  35^  ? 

6.  I  imported  500   yards   carpeting,    invoiced   at  $2   per 
yard.     What  will  be  the  duty  at  50%  of  the  value,  or 

ad  valorem  1 

7.  Mr.  A.  imported  5000  yards  sheeting.     What  will  be 
the  duty  at  4^  cents  per  yd.,  or  the  4^-  cents  specific  duty? 

DEFINITIONS. 

392.  Heal  Estate  is  fixed  property;  as,  houses,  lands, 
tenements,  etc. 

393.  Personal  Property  is  movable  property;    as 
money,  stocks,  mortgages,  cattle,  etc. 

394.  A  Tax  is  a  sum  of  money  assessed  upon  the  per- 
sons, property,  income,  or  business  of  individuals  for  public 
purposes. 


TAXES.  263 

395.  A  Property  Tax  is  a  tax  on  property.     It  is 
reckoned  at  a   certain  rate  per  cent,   on  the  estimated  or 
assessed  value  of  the  property. 

396.  A  Personal  Tax  is  a  tax  assessed  upon  the  per- 
son.    It  is  called  a  poll  or  capitation  tax. 

397.  An  Assessor  is  an  officer  appointed  to  estimate  the 
taxable  value  of  property  and  prepare  the  assessment  roll. 

398.  An  Assessment  Roll  is  a  list  of  the  names  of 
taxable  inhabitants,  and  the  value  of  each  person's  property. 

Before  taxes  are  assessed  a  complete  inventory  of  all  the 
taxable  property  must  be  made.  If  the  assessment  includes 
a  poll  tax,  then  a  complete  list  of  taxable  polls  must  be 
made  out. 

WRITTEN    EXERCISES. 

399.  1.  A  village  is  to  be  taxed  $3756  on  property  as- 
sessed at  $854315.     The  number  of  polls  at  $1.50  is  350. 
A's  property  is  assessed  at  $5000,  and  he  pays  5  polls;   B's 
property,  at  $7000,  and  he  pays  5  polls;    C's  property,  at 
$3425,  and  he  pays  3  polls.     What  will  be  the  tax  on  $1, 
and  what  will  be  the  tax  of  each? 

PROCESS. 

$1.50X350  =  $525,  Amount  of  poll  tax. 

$3756  — $525  =  $3231,  Amount  to  be  levied  on  property. 
$3231 --$854315  =  .00378,  or  3TV8o  mills  on  $1. 
$5000  X -00378=   $18.90,  A's  property  tax. 
$1.50X5          =   $  7.50,  A's  poll  tax. 
A's  entire  tax. 


$7000  X  .00378  =   $26.46,  B's  property  tax. 
$  1.50  X  5          =   $  7.50,  B's  poll  tax. 
$33.96,  B's  entire  tax. 


264 


PERCENTAGE. 


400.  A  table  like  the  following  is  commonly  made  by  those 
who  compute  the  taxes. 

ASSESSORS  TABLE.     (Rate,  .00378.) 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$.0037 

$  7 

$.026 

$40 

$.151 

$100 

$  .378 

$700 

$  2.646 

2 

.0074 

8 

.03 

50 

.19 

200 

.756 

800 

3.024 

3 

.011 

9 

.034 

60 

.23 

300 

1.124 

900 

3.402 

4 

.015 

10 

.038 

70 

.26 

400 

1.512 

1000 

3.78 

5 

.019 

20 

.076 

80 

.302 

500 

1.89 

2000 

7.56 

6 

.023 

30 

.111 

90 

.34 

600 

2.268 

3000 

11.34 

2.  Find  C's  tax  in  example  No.  1,  using  the  table. 

PROCESS. 

Tax  by  table  on  $3000  is  $11.34 
Tax  by  table  on  $  400  is  $  1.512 
Tax  by  table  on  $    20  is  $    .076 
Tax  by  table  on  $      5  is  $    .019 


1.50X3  = 


$12.947,  C's  property  tax. 
$  4.50  ,  C's  poll  tax. 
$17.447,  C's  entire  tax. 


RULE. — Divide  ike  sum  to  be  raised,  after  deducting  the  poll 
tax,  by  the  whole  amount  of  taxable  property,  and  the  quotient 
will  be  the  rate. 

Multiply  each  man's  property  by  the  rate,  and  to  the  product 
add  the  poll  tax,  if  any,  and  the  sum  will  be  the  whole  tax. 

3.  In  a  certain  town  the  assessed  value  of  property  was 
$1250500,  on  which  a  tax  was  levied  of  $5008.125.     The 
number  of  polls  was  425   at  $  .75   each.      A's  property  is  ; 
valued  at  $18500;   B's,  at  $22180;  C's,  at  15200.     A  pays 
5  polls;  B,  8  polls;  C,  9  polls.     What  is  the  tax  of  each? 


TAXES.  265 

4.  At  what  rate  must  $595000  worth  of  property  be  as- 
ssed  to  raise  a  tax  of  $2587,  if  there  are  300  polls  at  $1.50 
ch?     What  will  be  the  tax  on  property  valued  at  $3400? 

5.  In  a  town  containing  275  polls,  assessed  at  $1  each,  the 
assessment  roll  shows  that  the  taxable  property  is  valued  at 
$895970.     The  tax  is  $2310.90.     What  is  the  rate  of  taxa- 
tion? 

6.  A  tax  of  $35000  is  assessed  upon  a  certain  town.     The 
valuation  of  the  taxable  property  amounts  to  $4506000,  and 
there  are  650  taxable  polls  assessed  at  $1.25  each.     What 
will  be  the  tax  on  property  assessed  at  $11000? 

7.  What  sum  must  be  assessed  to  raise  $3750,  besides  pay- 
ing 2^  for  collection  ?     What  would  be  the  taxable  valuation 
of  property  to  raise  that'sum  if  the  rate  were  .003275? 


CASE  II. 

DUTIES   OR   CUSTOMS. 

401.  Duties,  or  Customs,  are  taxes  levied  by  gov- 
ernment upon  imported  goods. 

402.  A  Specific  Duty  is  a  fixed  tax  on  certain  articles 
without  regard  to  their  value. 

403.  An  Ad   Valorem  Duty  is  a  tax  of  a  certain 
per  cent,  on  the  net  value  of  the  articles  in  the  country  from 
which  they  have  been  brought. 

404.  Tare  is  the  allowance  made  for  the  weight  of  a 
box,  bag,  etc. 

405.  Leakage  and  Breakage  are  allowances  made 
br  leakage  and  breakage  during  transportation. 

406.  The  Duties,  or  Customs,  are  collected  by  officers  of 
lie  Government  in  the  Custom-houses. 


266  PERCENTAGE. 


WRITTEN   EXERCISES. 

407.    1.  What  is  the  duty  on  20  hhd.  molasses,  containing 
63  gal.  each,  at  9  cents  per  gal.,  allowing  5^  for  leakage? 

PROCESS.  ANALYSIS.— 

.,  0  £  A       -i  Since  5$,  is  allowed 

6  3  gal.  X    20  =  1260  gal.  gross.          fop  lea/age>  ^  of 

1260  gal  X  •  0  5  =  6  3  gal.  leakage.  the  gross  amount, 

1260  gal.  —    6  3  gal.  =  1  1  9  7  gal.  net.      or  63  gal.,  are  de- 
ft .09X1197  —  $107.73  duty.  ducted  for  leakage; 

and  since  the  duty 
on  1  gal.  is  9  cents,  on  1197  gal.  it  is  1197  times  9  cents,  or  $107.73. 

2.  What  is  the  duty,  at  5  cents  a  pound,  on  3750  pounds 
of  coffee,  allowing  5^  for  tare  ? 

3.  What  is  the  duty  on  500  pounds  of  raisins,  in  boxes, 
valued  at  10  cents  a  pound,  allowing  15^  for  tare,  when  the 
rate  of  duty  is  6^  ad  valorem  ? 

4.  What  will  be  the  duty  on  $3000  worth  of  merchandise 
if  the  rate  of  duty  is  15%  ad  valorem? 

5.  What  is  the  duty,  at  20^  ad  valorem,  on  7  tons  of  steel, 
of  2240  Ib.  each,  invoiced  at  17  cents  per  Ib.  ? 

6.  H.   K.  Thurber  &  Co.,  of  New  York,  imported  from 
Havana  75  hhd.  molasses,  63  gal.  each,  valued  at  35  cents 
per  gal. ;    125  hhd.  sugar,  500  Ibs.  each,  valued  at  6  cents 
per  Ib. ;    800  boxes  cigars,  valued  at  $8  per  box.     7^  was 
allowed  for  leakage  on  the  molasses;    duty  on  same,  25^; 
tare  on  sugar,  45  Ibs.  per  hhd. ;   duty  on  same,  30^ ;   duty 
on  cigars,  60^ .     What  was  the  amount  of  duties  paid  ? 

7.  A  wine  merchant  imported  45  casks  sherry  wine,  valued 
at  $65  per  cask;   56  casks  Madeira  wine,  valued  at  $60  per 
cask;  38  casks  German  wines,  valued  at  $37  per  cask.     If  an 
allowance  of  4^  be  made  for  leakage,  what  will  the  duty  be 
at 


408.    1.  Into  how  many  shares  can  $100000  capital  stock 
of  a  company  be  divided,  if  the  shares  are  $100  each? 

Shares  will  be  regarded  as  $100  each  unless  otherwise  specified. 

2.  How  much  of  the  capital,  or  capital  stock  does  a  man 
own  who  has  30  shares?     25  shares? 

3.  How  much  stock  is  represented  by  a  certificate  entitling 
the  holder  to  40  shares  ? 

4.  What  is  the  selling  price  or  market  value  of  10  shares  of 
railroad  stock,  when  stock  is  selling  at  its  original  value  or 
at  par  f 

5.  What  is  the  market  value  of  10  shares  of  stock,  if  it  is 
sold  at  105%  of  the  original  value  or  5%  above  parf 

6.  What  is  the  market  value  of  10  shares  of  stock,  if  it  is 
sold  at  95^  of  the  original  value  or  5%  below  parf 

7.  What  will  10  shares  of  stock  cost  at  5%  above  par,  if 
I  pay  a  dealer  in  stocks  or  stock-broker  \%  of  the  par  value 
of  the  stock  for  buying  it? 

8.  What  will  5  shares  of  stock  cost  at  5^  below  par,  if  I 
pay  a  broker  \%  for  buying  or  for  brokerage? 

9.  What  is  the  value  of  15  shares  of  bank  stock  at 
of  its  par  value? 

10.  What  is  the  cost  of  50  shares  Chicago  &  Eock  Island 
R.  R.  stock  at  90%  of  its  par  value? 

11.  What  is  the  cost  of  100  shares  Chicago  &  Alton  R  R. 
stock  at  95%  of  its  par  value? 

(267) 


268  PERCENTAGE. 

12.  What  is  the  cost  of  20  shares  Pacific  Mail  stock  at 
of  its  par  value  ? 

13.  A  company  whose   capital  stock  was  $50000,  gained 
$5000  above  expenses,  which  was  divided  among  the  stock- 
holders.   What  per  cent,  of  the  capital  stock  was  the  amount 
divided,  or  what  was  the  dividend? 

14.  What  amount  will  a  man  receive  who  owns  20  shares 
of  stock,  if  a  dividend  of  5^  is-  declared? 

15.  What  amount  will  a  man  receive  who  has  30  shares 
of  stock,  if  a  6^  dividend  is  declared  ? 

16.  A  company  whose  capital   stock  was  $50000,  lacked 
$5000  of  meeting  its  obligations.     What  per  cent,  of  the  stock 
was  the  deficiency? 

17.  If  the  deficiency  is  made  up  by  the  stockholders,  how 
much  will  be  the  assessment  on  a  stockholder  who  owns  20 
shares  of  the  above  stock? 

18.  How  much  will  be  the  assessment  on  15  shares  of  the 
above  stock? 

19.  What  will  be  the  annual  income  on  a  written  obliga- 
tion or  bond  for  $5000  which  yields  6^  interest  annually? 


DEFINITIONS. 

409.  A  Company  is  a  number  of  persons  associated 
together  for  the  prosecution  of  some  industrial  pursuit. 

410.  A   Corporate   Company  or   Corporation 

is  a  company  authorized  by  law  to  transact  business  as  an 
individual. 

411.  A  Charter  is  the  legal  document  which  defines  the 
rights  and  obligations  of  a  corporation. 

412.  Capital  Stock  is  the  property  or  money  invested 
in  the  business  of  the  company. 


STOCKS.  269 

413.  A.  Share  is  one  of  the  equal  divisions  of  the  capital 
stock  of  a  company. 

The  value  of  a  share  is  different  in  different  companies.  Unless 
otherwise  specified  it  will  be  regarded  as  $100. 

414.  A  Certificate  of  Stock  is  a  paper  issued  by  a 
corporate  company  giving  the  number  of  shares  to  which  the 
holder  is  entitled,  and  the  original  value  of  each. 

415.  Par  Value  is  the  value  named  in  the  certificate. 

When  shares  sell  for  the  value  named  on  their  face  the  stock  is  said 
to  be  at  par;  when  for  more  than  their  face,  above  par,  or  at  a  pre- 
mium; when  for  less  than  their  face,  below  par,  or  at  a  discount. 

416.  The  Market  Value  is  the  sum  for  which  stocks 
sell. 

417.  An  Installment  is  a  portion  of  the  capital  stock 
paid  by  the  stockholder. 

418.  A  Dividend  is  a  sum  divided  among  the  stock- 
holders as  the  profits  of  the  business. 

419.  An  Assessment  is  a  sum  which  the  stockhold- 
ers of  a  company  are  required  to  pay,  to  make  up  deficiencies 
or  losses. 

The  Government  and  Corporations  frequently  issue  Bonds 
for  the  purpose  of  raising  money. 

420.  A  Bond  is  a  written  obligation  under  seal,  securing 
the  payment  of  a  sum  of  money  on  or  before  a  specified  time. 

Interest  is  usually  paid  upon  bonds  at  fixed  dates,  as  an- 
nually or  semi-annually. 

421.  Coupons   are  certificates  of  interest  attached   to 
bonds.     They  are  cut  off  and  presented  for  payment  as  often 
as  the  interest  becomes  due. 


270  PERCENTAGE. 

422.  United   States    Government   Securities  are   of   two 
kinds,  viz:  bonds  which  are  to  be  paid  at  a  specified  time, 
and  bonds  which  are  to  be  paid  at  a  fixed  date,  or  some 
earlier  specified  time  at  the  option  of  the  Government. 

Bonds  are  sometimes  designated  by  combining  the  rate  of  interest 
and  the  time  of  payment.  Thus,  bonds  that  pay  6^?  interest  and  are 
payable  in  1881,  are  called  6's  of  '81. 

When  the  rate  is  uniform  for  a  class  of  bonds  it  is  omitted,  and 
the  time  of  redemption  and  payment  only  are  given.  Thus,  bonds 
that  may  be  redeemed  in  five  years,  and  are  payable  in  twenty  years, 
are  called  5-20's,  those  redeemable  in  ten  years  and  payable  in  forty 
years  are  called  10-40's. 

The  4<fa  consols  are  the  bonds  of  the  consolidated  loan.  They  are  re- 
deemable after  30  years  from  July  1,  1877. 

The  bonds  issued  by  States,  counties,  etc.,  are  named  from  the  rate 
of  interest  they  bear.  Thus,  New  Hampshire  bonds  that  bear  6<fc 
interest  are  called  New  Hampshire  6's. 

All  bonds  of  the  United  States  are  payable  in  coin  at  maturity. 

The  various  classes  of  United  States  bonds  are : 

1.  6's  of  '81      Interest  payable  semi-annually  in  gold. 

2.  5-20's,  issued  in  1862,  '64,  '65,  '67,  '68.     Interest  payable  semi- 
annually  in  gold  at  6^. 

3.  10-40's,  issued  in  1864.     Interest  payable  in  gold  semi-anuually, 
at  5$,  on  $500  and  $1000  bonds,  and  annually  on  $50  and  $100  bonds. 

4.  5's    of      '81.     Interest  payable  quarterly  in  gold. 

5.  4J's  of     '86.     Interest  payable  quarterly  in  gold. 

6.  4's    of  1901.     Interest  payable  quarterly  in  gold. 

7.  4^,    consols.     Interest  payable  quarterly  in  gold. 

423.  Stocks   is  a   term  which  includes  the  stock  of  a 
corporation,  the  various  U.  S.,  State,  and  county  bonds,  etc. 

424.  A  Stock-broker  is  a  person  whose  business  it  is 
to  buy  and  sell  stocks. 

425.  Brokerage  is  the  compensation  allowed  a  broker 
for  buying  and  selling  stocks. 


STOCKS.  271 

426.  The  computations  in  stocks  involve  the  same  elements 
as  the  fundamental  problems  in  Percentage.    The  correspond- 
ing terms  are: 

1.  The  Par  Value  of  the  stock  is  the  Base. 

2.  The  Rate  of  Premium,  or  Discount  is  the  Rate. 

3.  The  Premium  or  Discount  is  the  Percentage. 

4.  The  Market  Value  is  the  Amount,  or  Difference. 

427.  PRINCIPLE. — Brokerage  is  computed  on  the  par  value 
of  the  stock. 

EXERCISES. 

1.  What  is  the  cost  of  $1000  Delaware,  Lacka wanna,  and 
Western  R.  R.  bonds  at  108  when  \^0  additional  is  paid  for 
buying  ? 

2.  A  broker  received  $2510  to  invest  in  first  mortgage 
bonds  at  a  premium  of  25^.     If  he  charged  ^fo  brokerage, 
how  many  bonds  of  $100  each  did  he  buy? 

3.  A    broker   invested    $26000    in    quicksilver    stocks    at 
12^,    charging  \^0    brokerage.     How   many   shares   did  he 
buy? 

4.  A  man  invested   $5125  in  Union  Pacific  R.  R.  bonds 
at  $102,  paying  \<fo  brokerage.     How  many  bonds  of  $1000 
each  did  he  buy? 

5.  A  man  bought  a  number  of  shares  of  bank-stock  at 
125  and  sold  it  at  128,  gaining  $300.     How  many  shares 
did  he  buy? 

6.  Bought  bonds   at    115   and    sold  them   at  110,  losing 
$300.     How  many  bonds  of  $1000  each  did  I  buy? 

7.  How  much  must  I  invest  in  Missouri  State  bonds  at 
par,  which  pay  6^  interest  to  secure  an  income  of  $1200  an- 
nually ? 

8.  What  sum  must  I  invest  in  stock  at  par  paying  an 
annual  dividend  of  8^  to  realize  an  income  of  $4800  yearly? 


272  PERCENTAGE. 

9.  What  per  cent,  on  the  investment  does  6^  stock  yield 
if  it  is  bought  at  half  its  par  value  ?    If  at  \  of  its  par  value  ? 
10.  What  per  cent,  does  stock  pay,  if  it  yields  an  annual 
dividend  of  4^  and  is  bought  at  50^  of  its  par  value  ? 

11.  How  much  must  I  pay  a  share  for  stock,  which  yields 
an  annual  dividend  of  6^ ,  so  that  I  may  realize  12^  on  my 
money  annually?     18^?     24^?     30^? 

12.  For  what  price  must  bonds  bearing  9^   interest  be 
bought,  so  that  12^  may  be  realized  annually?     6^  ? 

13.  How  much  currency  can  be  obtained  for  $100  in  gold, 
when  gold  is  at  a  premium  of  4^  ?     9^  ?     12^  ? 

14.  How  much  currency  can  be  obtained  for  $100  in  gold, 
when  gold  is  at  a  premium  of  10^  ?     100^  ?     150^  ?     5^  ? 

15.  How  much  gold  at  6^  premium  can  I  buy  with  $106 
in  currency?     With  $318?     With  $159? 

16.  When  gold  is  at  a  premium  of  20^ ,  how  much  gold 
is  $1  in  currency  worth.     How  much  when  gold  is  at  50^ 
premium  ? 

17.  When  gold  is  selling  at  125,  what  is  the  value  in  gold 
of  a  United  States  greenback  for  $10? 

WRITTEN    EXERCISES. 

428.    1.  What  is   the  cost  of  500  shares   Delaware  and 
Hudson  Canal  Co.  stock  at  50^-,  brokerage  \^0  ? 

PROCESS.  ANALYSIS.  —  Since 

501^  4-10;  —50 3^  SOJ^of  the  par  value 

*/*•           4/*  of   the   stock   is   the 

50f^  of  $100  =  $50. 75,  cost  of  1  share.  price  paid  for  it,  the 

$50. 75X500 ^$25375,  the  entire  cost,      entire    cost    of    the 

stock,  including  the 

rate  for  brokerage,  is  50|^  of  the  par  value  of  the  stock.  And,  since 
the  par  value  of  a  share  of  the  stock  is  $100,  the  cost  of  a  share  will 
be  50f  ^  of  $100,  or  $50.75,  and  the  cost  of  500  shares  of  the  stock 
will  therefore  be  500  times  $50.75,  or  $25375. 


STOCKS.  273 

RULE. — Since  the  same  elements  are  involved  as  in  the  funda- 
l  problems  in  Percentage,  the  rules  are  the  same. 

FORMULAS. 

1.  Premium  or  Discount  =  Par  Value  X  Rate. 

2.  Rate  =  Premium  or  Discount  -f-  Par  Value. 

3.  Par  Value  =  Premium  or  Discount  -f-  Rate. 

4.  Par  Value  =  Market  Value  -*-  y  V- 

(  (1  —  Rate). 

*     nr    i  4.  IT  7          z>     -rr  /      f  +  Premium. 

5.  Market  Value  =  Par  Value  <        _. 

(  — Discount. 

2.  Find  the  cost  of  125  shares  Union  Pacific  R.  R.  stock, 
at  68|,  brokerage  ^  ? 

3.  What  will  $8000  U.  S.  5-20's,  coupon  bonds  of  '65,  cost 
at  108J,  brokerage  \<fr  ? 

4.  How  much  will  55  shares  C.  C.  C.  &  I.  R.  R.  stock 
cost  at  28f ,  brokerage  %fc  ? 

5.  What  must  be  paid  for  $5000  U.  S.   10-40's.  at 
premium,  brokerage  \^0  ? 

6.  Bought  35   shares  N.  Y.  C.  &  H.  R.  R.  R.  stock  at 
86^-,  and  sold  them  at  8ffl  advance.     How  much  did  I  gain? 

7.  Sold  135  shares  railroad  stock  at  a  discount  of 
paying  \^0  brokerage.     How  much  did  I  receive  for  it? 

8.  How  many  shares  of  bank-stock   at  5^  discount  can 
be  purchased  for  $3810,  if  \%  is  paid  for  brokerage? 

PROCESS.  ANALYSIS.— 

Ari^  Since  the  stock  was 

95i^         bought  at  5^  dis- 
or  40  shares,     count  it  was  bought 
at   95^  of  its  par 

value,  but  the  brokerage  increased  the  cost  J^,  so  that  each  dollar's 
worth  of  stock  cost  95^  of  its  par  value,  or  $  .95 \.  Therefore,  as 
many  dollars'  worth  can  be  bought  for  $3810  as  $  .95^  is  contained 
times  in  $3810,  which^is  4000  times,  or  40  shares  can  be  bought. 


274  PERCENTAGE. 

9.  Find  the  number  of  shares  of  E.  E.  .stock  at  102f,  which 
can  be  bought  for  $2575,  brokerage  f-%. 

10.  How  many  shares  of  N.  Y.  C.  &  H.  E.  E.  E.  stock 
at  9Sf,  can  be  bought  for  $28710,  brokerage  \%*i 

11.  How  many  shares  of  C.  B.  &  Q.  E.  E.  stock  at  109$,. 
can  be  bought  for  $66075,  brokerage  \%  ? 

12.  How  many  shares  of  Hartland  Ferry  stock  at  4%  dis- 
count, can  be  bought  for  $3330.25,  brokerage  \%  ? 

13.  How  many  shares  of  E.   E.   stock  at   3%    discount, 
can  be  bought  for  $2150.50,  brokerage  \%1 

14.  What  income  will  be  realized  from  investing  $4196.25 
in  5%  stock  purchased  at  93,  allowing  \%  for  brokerage? 

PROCESS.  ANALYSIS.— 


$4500X-05  —  $225,  annual  income.  its  par  value, 

every    dollar's 

worth  cost  $  .93  J  ;  and  as  many  dollars'  worth  can  be  bought  for 
$4196.25  as  $.93J  is  contained  times  in  that  sum,  which  is  4500 
times;  and  since  the  stock  paid  5^>  income,  the  entire  income  from 
$4500  is  6f0  of  $4500,  which  is  $225. 

15.  How  much  income  will  I  receive  annually  by  investing 
$1299  in  6%   stock  purchased  at  37%,   allowing  \%   bro- 
kerage? 

16.  What  will  be  the  income  from  investing  $4696.25  in 
Crawford  Co.  6's  at  45,  brokerage  \%. 

17.  Which  is  more  profitable,  and  how  much,  to  invest 
$5000  in  6^  stock  purchased  at  75%,  or  5%  stock  purchased 
at  60%? 

18.  U.  S.  5-20's  pay  6%  interest  in  gold.     What  will  be 
my  income  in  currency  by  investing  $11212.50  at  H2|-,  when 
gold  is  quoted  at  6|-%  premium? 

19.  Which  is  more  profitable,  to  buy  6%  bonds,  purchased 
at  90,  interest  payable  in  currency;  or  5%  bonds,  purchased 


STOCKS.  275 


at  95,  interest  payable  in  gold,  when  gold  is  quoted  at 
premium?     How  much  more  profitable  in  currency  is  it  on 
each  $100  invested? 

20.  A.  B.  Howard  sold  a  mill  for  $13850,  which  had  been 
paying  an  annual  profit  of  5^  of  that  sum,  and  invested 
the  proceeds  in  U.  S.  10-40's  at  lllf,  paying  \<fa  brokerage. 
Was  his  yearly  income  increased  or  diminished,   and  how 
much  in  currency,  gold  being  at  5%  premium? 

21.  How  much  must  be  invested  in  '6%  stock,  purchased 
at  90,  to  secure  to  the  purchaser  an  income  of  $900  annually? 

PROCESS.  ANALYSIS. 

$900--.  06  =--$15000,  par  value  of  stock.  JjJJ  Qf  ^  "£ 
$15000X.90  =  $13500,  cost  of  stock.  $.06,  it  will 

require      as 

many  dollars  to  secure  an  income  of  $900  as  $  .06  is  contained  times 
in  $900,  which  is  15000  times;  and  since  the  stock  is  selling  at  90^ 
of  its  par  value,  90$,  of  $15000,  which  is  $13500,  will  be  the  cost. 


22.  I  desire  to  invest  in  Ohio  &  Mississippi  E.  E.  bonds 
which  bear  6%  interest,  a  sum  of  money  sufficient  to  bring 
an  income  of  81000.     If  the  bonds  can  be  bought  at  91%, 
how  much  money  must  I  invest,  brokerage  \^0  ? 

23.  What  sum  must  I  invest  in  Louisiana  7's  at  107^  to 
secure  an  annual  income  of  $1750? 

24.  What  sum  must  I  invest  in  U.  S.  6's  at  112|  to  se- 
cure an  annual  income  of  $1750? 

25.  What  sum  must  be  invested  in  6%  stock  at  $84.50 
per  share  to  yield  an  income  of  $900  annually  ? 

26.  What  per  cent,  income  on  my  investment  will  I  re- 
ceive if  I  buy  6%  stock  at  20%  premium? 

PROCESS.  ANALYSIS. — Since  $1   of 

$  .  0  6  -T-  $  1 .  20  = .  0  5,  or  5%.       the  ;stock  costs  *L?°>  and 

'  the  income  from  it  is  $  .06, 

the  income  is  Tf^,  or  210-,  or  5^>  of  the  amount  of  the  investment. 


276  PEECENTAGE. 

27.  What  is  the  rate  per  cent,  of  income  from  bonds  which 
pay  1%  interest  when  they  are  bought  at  105? 

28.  If  stock  which  pays  a  semi-annual  dividend  of  5^% 
be  bought  at  10%  premium,  what  rate  per  cent,  of  income 
does  it  pay? 

29.  Which  affords  the  greater  per  cent,  of  income,  bonds 
bought  at  125  which  pay  8%,  or  bonds  which  pay  6%  bought 
at  a  discount  of  10^  ? 

30.  Which  is  more  profitable,  and  how  much  per  cent.,  to 
buy  New  York  7's  at  105^,  or  Louisiana  6's  at  98^? 

31.  What  per  cent,  of  income  does  stock  paying  9%  divi- 
dends afford  if  it  is  bought  at  112? 

32.  How  much  must  I  pay  for  New  York  6's  so  that  I 
may  realize  an  income  of  9^  on  the  investment? 

PROCESS.  ANALYSIS. — Since  the  income  is  6^> 

$.06-=-$. 09  =  . 66  2         °f  every  dollar  of  the  par  value  of  the 

stock,  if  an  income  of  9^  on  an  invest- 
ment be  desired,  then  6^  of  the  par  value  of  the  stock  must  be  9^  of 
the  sum  paid  for  $1  of  the  stock,  which  is  $  .66f ,  or,  the  stock  must 
be  bought  for  66f  <f0  of  its  par  value. 

33.  How  much  must  I  pay  for  stock  which  pays  a  divi- 
dend of  15^  so  that  I  may  realize  7^  on  my  investment? 

34.  How  much  premium  must  I  pay  on  stock  which  pays 
a  10%  dividend  so  that  I  may  realize  8^  on  my  investment? 

35.  At  what  price  must  I  buy  7%  stocks  so  that  they  may 
yield  an  income  equivalent  to  10%  stocks  at  par? 

36.  What  must  I  pay  for  New  York  6's  so  that  my  pur- 
chase may  yield  me  7%  ? 

37.  At  what  price  must  I  purchase  15^  stock  that  it  may 
yield  the  same  income  as  6^  stock  purchased  at  90? 

38.  What  is  the  currency  value  of  $9280  in  gold,  when 
gold  is  selling  at  107-|? 

39.  What  is  the  currency  value  of  $7225  in  gold,  when 
gold  is  at  a  premium  of 


INSURANCE.  277 

40.  What  is  the  value  in  gold  of  $5000  in  currency,  when 
fie  premium  on  gold  is  6%fo  ? 

41.  When  gold  is  selling  at  105 J,  what  is  the  value  of 
$7250  in  currency? 

42.  The  net  earnings  of  a  company  whose  capital  stock  is 
$2000000  was  $135000.     If  they  reserve  $5000  as  a  surplus 
fund,  what  per  cent,  dividend  can  they  declare? 

43.  Mr.  A.    purchased   250  shares  of  stock  at  75,   in   a 
company  whose  capital  was  $1500000.     The  gross  earnings 
of  the  company  for  1876  were  $225000,  the  expenses  were 
40%;  of  the  gross  earnings,  and  they  reserved  a  surplus  fund 
of  $10000.     What  per  cent,  dividend  did  Mr.  A.  receive  on 
his  investment? 

44.  A  capitalist  owning  200  shares  of  stock  of  $150  per 
share,  on  which  he  was  receiving  a  dividend  of  3%  semi- 
annually,  exchanged  them  for  6^  bonds  purchased  at  98. 
Did  he  gain  or  lose,  and  how  much  annually? 

45.  If  a  man  who  had  $5000  of  U.  S.  6's  of  '81  should  sell 
them  at  115,  and  invest  in  U.  S.  10-40's  purchased  at  105, 
would  he  gain  or  lose,  and  how  much  annually? 


INSURANCE. 

429.  1.  How  much  must  I  pay  to  secure  myself  against 
loss  by  fire,  or,  insure  my  property  for  $5000,  if  an  annual 
sum  or  premium  of  1^  is  charged  by  those  who  take  the  risk? 

2.  What  will  be  the  annual  premium  for  insuring  property 
for  $10000  at  \<f0  ? 

3.  What  will  be  the  annual  premium  for  insuring  property 
for  $1500  at  \\%  ? 

4.  What  is  the  premium  at  \°fo  for  insuring  a  vessel  and 
argo  to  the  amount  of  $36000? 

5.  What  will  it  cost  to  insure  $1500  worth  of  tea  at  f     ? 


278  PERCENTAGE. 

6.  What  must  be  paid  for  insuring  a  building  valued  at 
$3000,  for  |  of  its  value  at  \\<fa  ? 

7.  If  a  merchant  insured  his  goods  for  $2000  at  2^,  how 
much  premium  did  he  pay? 

8.  How  much  premium  did  a  merchant  pay  who  insured 
his  stock  of  boots  and  shoes  for  $6000  at  I$$? 

9.  A  man  paid  $25  premium  for  insuring  his  house  and 
furniture  against  loss  or  damage  by  fire.     For  how  much  was 
he  insured,  if  the  rate  of  insurance  was  \^0  ? 

10.  For  how  much  was  a  man  insured  who  paid  $50  pre- 
mium for  insuring  his  barn  and  live  stock  at  \%  ? 

11.  How  much  was   the  amount  of  insurance  when  the 
premium  paid  for  insuring  a  house  and  furniture  at  f  ^  was 


12.  For  how  much  is  a  farmer  insured  on  his  barns  and  the 
grain  in  them,  who  pays  $60  premium,  when  the  rate  is  2^  ? 

DEFINITIONS. 

4-30.  Tnsiircince  is  indemnity  against  loss  or  damage. 
It  is  of  two  kinds:  Property  Insurance  and  Personal  Insurance. 

x- 

CASE  I. 

PROPERTY  INSURANCE. 

431.  Property  Insurance  is  indemnity  against  loss 
or  damage  by  fire,  or  Fire  Insurance;  against  loss  or  damage 
by  casualties  at  sea,  or  Marine  Insurance;  and  against  loss  or 
damage  by  fire,  lightning,  etc.,  to  cattle,  horses,  etc.,  or  Live 
Stock  Insurance. 

432.  The  Policy  is  the  contract  or  agreement  between 
the  insurance  company  and  the  person  insured. 

433.  The  Premium  is  the  sum  paid  for  insurance. 


INSURANCE. 


436.  The  computations  in  insurance  involve  the  same  ele- 
ments as  do  the  fundamental  problems  in  Percentage.  The 
corresponding  terms  are: 

1.  The  Amount  Insured  is  the  Base. 

2.  The  Kate  of  Premium  is  the  Hate. 

3.  The  Premium  is  the  Percentage. 

WRITTEN     EXERCISES. 

1.  How  much  is  the  premium  on  a  policy  of  insurance  on 
a  dwelling  for  $3000  if  the  rate  of  premium  is  \\%  ? 

PROCESS.  ANALYSIS. — Since  the  pre- 

.011  =  837.50       mium  is  l*fi  of  the  amount 

insured,  to  find  the  premium, 
\\cjc  of  $3000  must  be  found,  which  is  $37.50. 

RULE. — Since  the  same  elements  are  involved  as  in  the  funda- 
mental problems  in  Percentage  the  rules  are  the  same. 

FORMULAS. 


1.  Sum  insured  X  Rate  =  Premium. 

2.  Premium  ~-  Sum  insured  =  Rate. 

3.  Premium  ~-  Rate  =  Sum  insured. 


280  PERCENTAGE. 

2.  How  much  is  the  premium  for  insuring  a  stock  of  goods 
for  $15000  at  \\<fr  ? 

3.  A  man  had  his  house  insured  for  $5000  paying  f^, 
and  his  furniture  for  $3000   paying  \<fa.     How  much  was 
the  premium? 

4.  How  much  must  be   paid  for  insuring  a  flouring  mill 
valued  at  $18000  for  f  of  its  value  at  2%fi  ? 

5.  A  vessel  valued  at  $80000,  with  its   cargo  valued  at 
$65000,  was  insured  for  f  of  its  value  at  1^.     What  was 
the  premium  ?     What  would  be  the  actual  loss  to  the  insur- 
ance company  if  the  above  vessel  should  be  lost  at  sea? 

6.  Property  was  insured  for  $15850  at  3^.    What  amount 
of  premium  was  paid  ? 

7.  Paid  $275  for  insuring  property  at  \^0 .     What  was  the 
amount  insured? 

8.  Paid    $325   for   insuring    property  valued   at   $16250. 
What  was  the  rate  of  premium  ? 

9.  A  insured  his  buildings  for  $9500,  paying  a  premium 
of  $47.50.     What  rate  did  he  pay? 

10.  Mr.  Orcott  paid  $175  for  insuring  his  block  of  mer- 
cantile buildings  at   1^^.      How  much    insurance   did  he 
procure  ? 

11.  Mr.  James  paid  $652.50  for  insuring  property  valued 
at  $43500.     What  was  the  rate? 

12.  A  man  paid  $180  for  insuring  his  saw  mill  for  f  its 
value  at  3^.     What  was  the  value  of  the  mill? 

13.  A  merchant  whose  stock  of  goods  was  valued  at  $30000 
insured  it  for  three-fourths  of  its  value  at  f^.     In  a  fire  he 
saved  $5000  of  the  goods.     What  was  his  loss  ?     What  was 
the  loss  of  the  insurance  companies? 

14.  The  price  of  a  quantity  of  silks  was   discovered  by 
knowing  that  they  were  insured  at  4|^   for  two-thirds  of 
their  value,  and  the  premium  paid  was  $400.     What  was  the 
price  of  the  silks? 


INSURANCE.  281 

15.  A  manufacturing  company  insured  its  works  for  f  of 
their  value  at  2^,  paying  as  premium  $1657.50.     What 
was  the  value  of  the  works? 

16.  A  vessel  and  cargo  were  insured  for  %  of  their  value 
at  H%.     The  premium  amounted  to  $2475.     At  what  price 
were  the  vessel  and  cargo  valued? 

17.  Paid  $225  for  insuring  a  store  and  its  contents  for  f  of 
their  value  at  1-^.     The  stock  was  worth  half  as  much  as 
the  store.     What  was  the  value  of  each? 

18.  For  how  much  must  a  block  of  stores  worth  $20000 
be  insured,  so  that  the  insurance  will  cover  f  of  the  value 
of  the  property  and  the  amount  of  the  premium  at 


CASE  II. 
PERSONAL  INSURANCE. 

4:37.  Personal  Insurance  is  indemnity  against  loss 
of  life,  or  Life  Insurance;  against  loss  occasioned  by  accidents, 
or  Accident  Insurance;  and  against  loss  occasioned  by  sickness, 
or  Health  Insurance. 

The  policies  issued  by  Life  Insurance  Companies  are  of 
various  kinds,  the  chief  of  which  are  the  Life  Policy  and  the 
Endowment  Policy. 

438.  A  Life  Policy  secures  a  sum  of  money  at  the 
death  of  the  person  insured. 

439.  An   Endowment   Policy   secures   a   sum    of 
money  at  a  specified  time,  or  at  death,  if  it  occur  before  the 
specified  time. 

440.  Accident  and  Health  Policies  secure  a  stip- 
ulated sum  for  a  certain  time,  in  case  of  a  disabling  accident 
or  sickness,  and  the  face  of  the  policy  in  case  of  death  by 
accident. 


282  PERCENTAGE. 

441.  The  rates  of  premium  are  based  upon  the  expectation 
of  life,  determined  by  observing  the  death  rate  per  thousand 
inhabitants. 


WRITTEN    EXERCISES. 

442.  1.  What  premium  must  be  paid  annually  for  a  life 
policy  of  $5000,  at  $21.10  per  $1000? 

ANALYSIS. — Since  the  rate  of 

PKOCESS.  premium  is  $21.10  annually  for 

$21.10X5  =  $105.50       WOO,  the  premium  for  $5000  is 

5  times  $21.10,  which  is  $105.50. 

2.  How  much  will  be  the  annual  premium  on  a  life  insur- 
ance policy  for  $3000  at  $31.30  per  $1000? 

3.  What  is  the  amount  of  annual  premium  on  a  life  policy 
for  $5500  at  $26.38  per  $1000? 

4.  If  a  person  who  is  insured  for  $5000,  at  an  annual  pre- 
mium of  $28.90  per  $1000,  dies  after  9  payments,  how  much 
more  will  his  heirs  get  than  has  been  paid  in  premiums? 

5.  If  a  man  insures  his  life  for  $5000,  paying  $22.90  per 
$1000,  and  dies  immediately  after  paying  his  annual  premium 
for  30  years,  what  is  the  result  of  the  investment,  reckoning 
simple  interest  at  6%  on  the  premiums  paid? 

60  If  Mr.  Bowditch  insures  his  life  for  $5000  on  the  endow- 
ment plan  when  he  is  40  years  of  age,  the  policy  to  be  payable 
when  he  is  55  years  of  age,  paying  therefor  $54.90  for  $1000, 
will  he  gain  or  lose  by  insuring,  reckoning  simple  interest  at 
1%  on  the  premiums  paid,  if  he  lives  till  the  policy  is  paid? 

7.  A  traveling  agent  has  an  accident  insurance  policy  for 
$3000,  for  which  he  pays  $50  per  year.  His  weekly  com- 
pensation, in  case  of  a  disabling  injury,  is  $30.  Immediately 
after  he  makes  his  fifteenth  payment  he  is  disabled  by  an  in- 
jury for  20  weeks.  Does  he  gain  or  lose  by  the  insurance, 
reckoning  simple  interest  at  6%  on  the  premiums  paid? 


443.  1.  When  A  owes  B  $500,  and  B  owes  A  $500,  how 
may  the  accounts  be  settled  without  any  transfer  of  money 
taking  place? 

2.  When  A  in  Chicago  owes  B  in  New  York  $500,  and  C 
in  New  York  owes  A  $1000,  how  can  A  pay  his  indebted- 
ness to  B  without  remitting  the  money? 

3.  What  will  be  the  indebtedness  of  A,  B  and  C  to  each 
other  after  the  transaction  has  taken  place  ? 

4.  A  and  C  live  in  the  same  city,  and  B  in  a  distant  city. 
A  owes  B  $2000,  and  B  owes  C  $1000.     How  may  B  pay 
his  indebtedness  to  C  without  remitting  the  money? 

5.  What  will  be  their  indebtedness  to  each  other  after  A 
has  paid  B's  order,  or  draft? 

6.  What  will  a  draft  for  $500  cost,  payable  when  it  is 
presented,  or  at  sight,  if  \%  premium  is  charged  for  it? 

7.  How  much  should  be  deducted  from  the  price  of  the 
above  draft  if  it  is  not  to  be  paid  until  two  months,  money 
being  worth  6%  ? 

8.  What  will  be  the  cost  of  a  draft  for  $50,  payable  at 
sight,  if  it  is  purchased  at  1%  discount? 

9.  What  will  be  the  cost  of  a  sight  draft  for  $300,  pur- 
chased at  \%  premium? 

10.  If  A  in  Nashville  owes  B  in  New  Orleans  $1000,  and 
C  in  New  Orleans  owes  D  in  Nashville  $1500,  how  may  A 
pay  his  indebtedness  without  remitting  the  money  ? 

(283) 


284  PERCENTAGE. 

11.  If  the  premium  is  |^,  how  much  will  it  cost  me  to 
remit  a  draft  for  $800  from  Cincinnati  to  Cleveland? 

12.  If  a  man  sells  a  draft  for  $500,  at  a  premium  of  f^, 
how  much  does  he  receive  for  it? 

13.  A  wishes  to  send  to  his  agent  in  New  Orleans  a  draft 
for  $5000.     If  the  premium  on  exchange  is  f^,  how  much 
will  the  draft  cost  him? 

14.  When  I  pay  $2025  for  a  sight  draft  on  New  York  for 
$2000,  what  is  the  premium,  or  rate  of  exchange? 

15.  When  I  can  buy  a  sight  draft  on  Chicago  for  $2000, 
paying  for  it  $1980,  what  is  the  rate  of  exchange? 

16.  If  Mr.  Burt  pays  $4975  for  a  sight  draft  on  Cincinnati 
for  $5000,  at  what  rate  is  exchange? 


DEFINITIONS. 

444.  Exchange  is  the  method  of  making  payments  in 
distant  places  without  transmitting  money. 

445.  A  Draft  or  Sill  of  Exchange  is  a  written 
order  by  one  person  to  another  to  pay  a  specified  sum  of 
money  to  the  person  named  in  the  writing  or  his  order. 

FORM   OF   A    DRAFT. 

$384 T2oV  CINCINNATI,  O.,  July  20,  1877. 

Twenty  days  after  sight  pay  to  the  order  of  the  First  Na- 
tional Bank,  Chicago,  111.,  Three  Hundred  Eighty -four  -f-fa 
Dollars,  value  received,  and  charge  to  the  account  of 

To  JAMES  H.  HOOSE  &  Co.,  J°NES  BROS.  &  Co. 

Chicago,  111. 

There  are  primarily  three  parties  connected  with  a  draft,  viz : 
the  person  who  signs  it,  the  person  who  is  ordered  to  pay  the 
money,  and  the  person  to  whom  the  money  is  to  be  paid. 


EXCHANGE.  285 

446.  The  Drawer  is  the  person  who  signs  the  draft. 

447.  The  Drawee  is  the  person  who  is  ordered  to  pay 
the  money. 

448.  The  Payee  is  the  person  to  whom  it  is  ordered 
that  the  money  be  paid. 

449.  A  Sight  Draft  is  one  which  is^  to  be  paid  when 
it  is  presented  to  the  drawee. 

450.  A  Time  Draft  is  one  payable  at  a  specified  time, 
after  its  presentation  to  the  drawee,  or  after  date. 

On  Time  Drafts  three  days  of  grace  are  usually  allowed. 

451.  Accepting  a  draft  is  agreeing  to  pay  it  when  it 
is  due.     This  is  done   by  the   drawee  writing    "  Accepted" 
across  the  face  of  the  draft,  .with  his  name  and  the  date. 

Exchange  is  of  two  kinds,  viz :  Domestic  or  Inland, 
and  Foreign. 

CASE  I. 

DOMESTIC   EXCHANGE. 

452.  Domestic  Exchange  treats  of  drafts  payable 
in  the  country  where  they  are  made. 

WRITTEN    EXERCISES. 

453.  1.  What  will  be  the  cost  of  procuring  a  sight  draft 
on  New  York  for  $5000  at  \%  premium? 

PKOCESS.  ANALYSIS. — Since  exchange 

«M  4_<fc    001—1    OOi  on  New  York  is  at  *#  Pre- 

••.  o-o  } =  i . uo *        mium>  every  $1  of  the  draft 

$1.00^X5000 --^$5025       will  cost  $1.00},  and  a  draft 

for  $5000  will   therefore  cost 
the  purchaser  5000  times  $1,00},  which  is  $5025. 


286  PEKCENTAGE. 

2.  What  will  be  the  cost  in  New  Orleans  of  a  draft  on 
New  York,  payable  60  days  after  sight,  for  $5000,  exchange 
being  at  \\%  premium  ?  ANALYsis.-Since  the 

-    PROCESS.  exchange  on  New  York 

<fc1     J_ft    01  ^  —  «M     01  ^  is    at    1J^    premium, 

$1.  +  $.015-       1.U15  every  $1  /of  the  draft 

61.015  —  $.008f  =  *1.006±        would  cost  $1.015  if 
$1.006JX5000  =  $5031.25        paidatsight.    But  since 

it  is  not  to  be  paid  in 

New  York  for  63  days,  the  banker  in  New  Orleans  who  has  the  use 
of  the  money  for  that  time,  since  he  is  not  obliged  to  pay  the  money 
in  New  York  for  63  days,  allows  the  bank  discount  on  the  face  of  the 
draft  for  that  time.  The  bank  discount  on  $1,  at  the  legal  rate  in  the 
State  of  Louisiana,  for  the  given  time,  is  $.008f,  which,  subtracted 
from  $1.015,  gives  $1.006},  the  cost  of  $1  of  the  draft.  Since  the  cost 
of  a  draft  for  $1  is  $1.006 J,  the  cost  of  a  draft  of  $5000  will  be  5000 
times  $1.006 J,  or  $5031.25. 

3.  What  will  be  the  cost  in  Memphis,  Tenn.,  of  a  sight 
draft  on  Cincinnati  for  $1000,  the  rate  of  exchange  being 
\<fi)   premium? 

4.  If  exchange  on*  Chicago  is  1J^£  premium,  what  will  be 
the  cost  in  Savannah,  Ga.,  of  a  sight  draft  for  $3000? 

5.  A  merchant  in  Chicago  bought  a  draft  on  New  York 
for  $5000,  payable  30  days  after  sight.     What  did  it  cost 
if  exchange  was  \^0  premium  ? 

6.  What  will  be  the  cost  in  Buffalo,  N.  Y.,  of  a  draft  for 
$1500  on  Cleveland,  O.,  payable  90  days  after  date,  when 
exchange  is  \<f0  discount? 

7.  How  much  will  be  realized  from  the  sale  of  a  draft  for 
$5000  at  \%  discount? 

8.  How  much  will  be  realized  from  the  sale  of  a  draft  for 
$3000  sold  at  \<f0  premium? 

9.  When  exchange  is  at  \<fa  premium,  what  will  be  the 
cost  of  a  draft  for  $5000,  purchased  in  Chicago  pn  Omaha, 
to  be  paid  90  days  after  date? 


EXCHANGE.  287 

10.  If  exchange  is  at  \<fa  premium,  what  will  a  draft  for 
$1500  cost,  purchased  in  St.  Paul,  Minn.,  on  Dayton,  0., 
payable  in  60  days,  without  grace? 

11.  If  exchange  is  at  ^fo  premium,  what  will  a  draft  for 
$5000  on  New  York  cost  in  Cincinnati,  payable   60  days 
after  date? 

12.  How  large  a  draft  on  New  Orleans  can  be  purchased 
for  $5000,  when  exchange  is  at  \\%  premium? 

PROCESS  ANALYSIS.  —  Since   ex- 

«1  j  ft  m  i      <iu   01  ^          change  is  at  ltf°  pre" 
I  +  f  .  0  1    =      1.015  it  wm  ^  $LQ15 


$5000-f-1.015  =  $4926.11       to  buy  a  draft  for  $1,  and 

$5000  will  buy  a  draft  for 
as  many  dollars  as  $1.015  is  contained  times  in  $5000,  or  $4926.11. 

13.  How  large  a  draft  on  Washington,   D.  C.,   payable 
60  days  after  sight,  can  be  bought  in  Nashville,  Tenn.,  for 
$3000,  when  exchange  is  at  \°fo  discount? 

PROCESS.  ANALYSIS.  —  Since    ex- 

<M        <fc    m        ft    QQ  change  is  at  14,  discount, 

*'U  it  would  cost  $.99  to  buy 

$.99  —  $.0105:=$.  97  9  5  a  draft  for  $1  if  it  were 

$3000-=-.9795  =  $3062.78        Payable  at  sight  ;  but  since 

the  draft  is  not  to  be  paid 

until  63  days,  the  banker  in  Nashville  who  has  the  use  of  the  money 
for  63  days,  allows  bank  discount  on  the  face  of  the  draft  for  that 
time,  or  $  .0105  for  every  dollar.  Therefore  since  it  costs  $  .9795  to 
purchase  a  draft  for  $1,  $3000  will  purchase  a  draft  for  as  many 
dollars  as  $  .9795  is  contained  times  in  $3000,  or  $3062.78. 

14.  How  large  a  sight  draft  can  be  purchased  on  Chicago 
for  $5725,  when  the  rate  of  exchange  is  \fo  premium? 

15.  What  will  be  the  face  of  a  30-day  draft  purchased  for 
$1500,  if  the  rate  of  exchange  is  \*cfc  premium  and  the  rate 
of  discount  is  6%  ? 

16.  If  I  pay  $1200  for  a  draft  payable  in  60  days,  when 


288  PERCENTAGE. 

the  premium  on  exchange  is  |^,  and  the  rate  of  discount 
is  9^J ,  what  will  be  the  face  of  the  draft  ? 

17.  How  large  a  sight  draft  on  New  York  can  be  pur- 
chased in  Chicago  for  $10000,  if  exchange  is  \^0  discount? 

18.  A  commission  merchant  in  St.  Louis,  Mo.,  sold  goods 
amounting  to  $3500  for  a  man  in  Denver,  Col.     He  sent  the 
amount  due  by  a  draft  payable  in  30  days  after  sight,  ex- 
change being  \^0  premium.     How  large  a  draft  did  he  pur- 
chase ? 

19.  How  large  a  draft  at  sight  on  San  Francisco  can  I 
purchase  for  $1750  if  exchange  is  at  \<fa  premium? 

CASE  II. 
FOEEIGN   EXCHANGE. 

454.  Foreign  Exchange  treats  of  drafts  made  in 
one  country  and  payable  in  another. 

Foreign  bills  of  exchange  in  the  United  States  are  drawn  on  London, 
Paris,  Berlin,  Antwerp,  Amsterdam,  Hamburg,  Bremen,  and  other  com- 
mercial centers;  but  drafts  on  London  and  Paris  are  more  common, 
inasmuch  as  they  are  paid  anywhere  on  the  continent  of  Europe. 

455.  A  Set  of  Exchange  consists  of  three  drafts  or 
bills  of  the  same  date  and  tenor,   named,  respectively,  the 
first,  second,  and  third  of  exchange.     They  are  sent  by  differ- 
ent mails,  so  that  if  one  is  lost  another  may  be  presented. 
When  one  bill  of  the  set  is  paid  the  others  are  void. 

456.  The  value  of  a  pound  sterling  previous  to  1873  was 
fixed  at  $4.44-|.     In  1873  Congress  fixed  the  value  of  the 
sovereign  in  U.  S.  gold  coin  at  $4.8665,  which  is  now  the  par 
of  ex-change. 

The  value  of  a  franc  is  about  $  .193,  or  about  5.18  francs 
to  one  dollar  in  gold.  Exchange  on  Paris  is  quoted  at  a  cer- 
tain number  of  francs  to  a  dollar  in  gold. 


EXCHANGE.  289 


WRITTEN    EXERCISES. 

457.  1.  What  is  the  cost  in  currency,  in  New  York,  of  a 
sight  draft  on  London  for  £312  15s.  5d.,  when  exchange  is 
$4.87  for  a  pound  sterling  and  gold  at  106. 

ANALYSIS. 

£312  15s.  5d.  =  £312.7708,  value  in  pounds  and  decimals  of  a  pound. 
$4.87  X  312.7708  =$1523.193  +,  the  cost  in  U.  S.  gold. 
$1523.193  X  1.06  =  $1614.59,  the  cost  in  U.  S.  currency. 

2.  How  large  a  bill  of  exchange  at  sight  on  London  can 
be  bought  in  New  York  for  $2984.38  in  currency,  exchange 
being  at  $4.86  for  a  pound  sterling  and  gold  at  107^-? 

ANALYSIS. 

$2984.38  -*-$1.07J  =  $2776.167,  value  of  currency  in  gold. 
$2776.167-1- $4.86  =£571.2263+,  value  in  pounds  sterling. 
£571.2263  =  £571  4s.  6Jd.,-the  face  of  the  draft. 

3.  How  large  a  bill  of  exchange  at  sight  on  London  can 
be  bought  in  New  York  for  $3762.50  in  currency,  when  gold 
is  at  105 J  and  exchange  is  at  $4.87? 

4.  What  will  be  the  face  of  a  sight  draft  on  London, 
which  is  purchased  in  Philadelphia  for  $5928.75  in  currency, 
when  gold  is  at  106^  and  exchange  is  at  $4.85|-? 

5.  What  will  be  the  face  of  a   sight   draft  on  London, 
which  is  purchased  in  Norfolk,  Va.,   for  $5575.20  in  cur- 
rency, when  exchange  is  at  $4.87y,  and  gold  is  selling  at 
107|? 

6.  What  must  be  paid  in  currency  for  a  bill  of  exchange 
on  Paris,  at  sight,  for  3269  francs,  exchange  being  at  5.15 
francs  to  the  dollar,  and  gold  at  105f  ? 

7.  What  must  be  paid  in  currency  for  a  sight  bill  of  ex- 
change on  Paris  for  8950   francs,  exchange  being  at  5.19 
francs  for  one  dollar,  and  gold  at  106^? 

19 


290  AVERAGE    OF    PAYMENTS. 

8.  What  will  be  the  face  of  a  sight  draft  on  Paris  which  is 
bought  in  Baltimore  for  SI 575  in  currency,  when  the  rate  of 
exchange  is  5.19  francs  for  a  dollar,  and  gold  is  at  107^? 

9.  An  American  bought  a  sight  draft  on  Paris  for  5725 
francs.     What  was  the  currency  value  of  the  draft  when  ex- 
change was  at  5.20  francs  for  a  dollar  and  gold  was  106^? 

10.  What  is  the  value  in  currency  of  a  bill  of  exchange, 
at  sight,  on  London,  for  £895  10s.,  when  exchange  is  $4.87 
for  a  pound  sterling  and  gold  is  quoted  at  106f  ? 

11.  An  American  in  Philadelphia  purchased  a  sight  draft 
on  London  for  £585  10s.  5d.     What  was  the  currency  value 
of  the  draft,  if  exchange  was  at  par  and  gold  at  107^-? 

12.  What  will  be  the  cost,  in  currency,  of  a  sight  bill  of 
exchange  on  London  for   £875   5s.    4d.,   when  exchange  is 
$4.88-^-  for  a  pound  sterling  and  gold  is  quoted  at  104^? 


AVERAGE  OF  PAYMENTS. 

458.    1.  How  long  may  $1  be  kept  to  balance  the  use  of 
$5  for  2  months? 

2.  How  long  may  $1  be  kept  to  balance  the  use  of  $7  for 

3  months? 

3.  How  long  may  $10  be  kept  to  balance  the  use  of  $5  for 

2  months? 

4.  How  long  may  $20  be  kept  to  balance  the  use  of  $10  for 

4  months? 

5.  How  long  may  $50  be  kept  to  balance  the  use  of  $25  for 
7  months? 

6.  How  long  may  $40  be  kept  to  balance  the  use  of  $80  for 

3  months? 

7.  I  owe   B   two  debts  of  equal   amount,   one  due   in   3 
months  and  the  other  in  6  months.     When  may  I  pay  both 
at  one  payment? 


AVERAGE   OF    PAYMENTS.  291 

8.  I  owe  A  two  debts  of  $20  each,  one  due  in  2  months 
and  the  other  in  4  months.     When  may  I  pay  both  at  one 
payment  ? 

9.  If  I  pay  $20  three  months  before  it  is  due,  how  long 
after  it  is  due  may  I  keep  $30  to  balance  it? 

10.  If  I  owe  $20,  due  in  4  months,  and  $40,  due  in  6 
months,  at  what  time  can  I  equitably  pay  both  debts  at  one 
payment? 

11.  If  I  owe  $30  due  in  3  months,  and  $10  due  in  7 
months,  when  may  I  equitably  pay  both  debts  by  a  single 
payment  of  $40? 

DEFINITIONS. 

459.  Averaging  Payments  is  finding  the  equitable 
time  for  discharging,  by  one  payment,  sums  due  at  different 
times. 

460.  The  Average  Time  is  the  date  at  which  the  debts 
may  be  equitably  discharged  by  a  single  payment. 

461.  The  Term  of  Credit  is  the  time  that  must  elapse 
before  the  debt  becomes  due. 

462.  The  Average  Term  of  Credit  is  the  time  that 
must  elapse  before  the  debts  due  at  different  times  may  be 
equitably  discharged  by  a  single  payment. 


CASE  I. 

463.  When  the  terms  of  credit  hegin  at  the  same 
date. 

1.  A.  T.  Stewart  &  Co.  sold  a  bill  of  goods  upon  the  fol- 
lowing terms :  $400  cash,  $300  due  in  2  months,  and  $400 
due  in  4  months.  At  what  time  might  the  whole  indebted- 
ness be  equitably  discharged  by  a  cash  payment? 


292  AVERAGE   OF    PAYMENTS. 

PROCESS.  ANALYSIS. — Since 

$400  for    0  mo.  =  $1  for .  ?400  was  to  be  Paid 

OAA    f  &1     f  Gf\f\  m      C!lSh>      there      W£IS 

300  for     2  mo.  =  $1  for     600  mo.  no  term  of  credit  for 

400  for    4  mo.--=$l  for  1600  mo.  that  sum.    gince 

$1100  2200  mo.  $300  was  to  be  paid 

in  2  mo.,  the  use  of 

2200  mo. -f-1 100— 2  mo.    Average  term  of  credit,  that  sum   for  2  mo. 

is  equal  to  the  use 

of  $1  for  600  mo.,  and  the  use  of  $400  for  4  mo.  is  equal  to  the  use 
of  $1  for  1600  months.  Hence,  the  credit  of  the  whole  debt,  $1100, 
is  equal  to  the  credit  of  $1  for  2200  mo.,  or  $1100  for  TTVo  part  of 
2200  mo.,  which  is  2  months,  the  average  term  of  credit. 

RULE. — Multiply  each  debt  by  its  term  of  credit,  and  divide 
the  sum  of  the  products  by  the  sum  of  the  debts.  The  quotient 
will  be  the  average  term  of  credit. 

2.  H.  B.  Claflin  &  Co.  sold  a  bill  of  goods  amounting  to 
$2300,  on  the  following  terms:    $300  cash,  $1200  due  in  3 
months,  and  the  balance  due  in  4  months.     What  was  the 
average  term  of  credit. 

3.  Field,  Leiter  &  Co.  sold  a  bill  of  goods  payable  as  fol- 
lows: $500  in  1  month,  $500  in  2  months,  and  $800  in  4 
months.     What  was  the  average  term  of  credit? 

4.  Whitney  &  Co.  sold  a  bill  of  lumber  on  the  following 
terms:    $1500  cash,  $3000  payable  in  30  days,  and  $2000 
payable  in  90  days.     At  what  time  will  the  debt  be  payable 
in  one  cash  payment? 

5.  H.  K.  Thurber  &  Co.  sold  to  F.  N.  Burt  a  bill  of  goods 
amounting  to  $2400,  payable  as  follows :  ^  in  30  days,  \  the 
remainder  in  60  days,  and  the  balance  in  4  months.     What 
was  the  average  term  of  credit? 

6.  Mr.  Birge  bought  a  bill  of  goods  amounting  to  $3000, 
payable  as  follows:   \  in  3  mo.,  \  in  2  mo.,  and  the  rest  in 
4  mo.     What  was  the  average  term  of  credit  ? 


AVERAGE   OF   PAYMENTS.  293 


CASE  II. 

464.  When  the  terms  of  credit  begin  at  different 
dates. 

1.  Find  the  average  time  of  payment  of  the  following 
bills:  Feb.  10,  1877,  $400  due  in  2  mo.;  March  15,  1877, 
$350  due  in  3  mo.  ;  and  April  12,  1877,  $300  due  in  3  mo. 

PROCESS.  ANALYSIS.  - 

<IJ».AA   -,         A      .-,  1A  Adding  to    the 

$400  due  April  10.     400  date  of  the  pur- 

350'  due  June  15.     350x66  =  23100         chase   of   each 
300  due  July    12.     300  X  93  =  27900         bill  its  term  of 


1050  51000         cri>  we   o~ 

tain    the    time 

51000  -*-  1050  =  48^f  days.  when  it  is  due, 

April  10  +  49  days  =  May  29,  average  term,     and  so  we  have 

$400  due  April 

10,  $350  due  June  15,  $300  due  July  12.  The  average  time  when 
the  hills  will  he  due  will  be  either  after  the  earliest  date,  or  before 
the  latest  date,  and  so  we  may  select  either  of  these  dates  from  which 
to  compute  the  average  time.  Selecting  the  earliest  date,  we  find  that 
$350  was  due  66  days  after  that  time,  and  $300  was  due  93  days 
thereafter.  Averaging,  as  in  Case  I,  we  find  the  term  of  credit  to  be 
48Jf,  or  a  fraction  more  than  48  days,  which  must  be  49  days.  This, 
added  to  April  10,  gives  May  29,  the  average  time  of  payment. 

RULE.  —  Select  the  earliest  date  at  which  any  debt  becomes  due 
for  the  standard  date,  and  find  how  long  after  that  date  the  other 
amounts  become  due. 

Find  the  average  term  of  credit  by  multiplying  each  debt  by 
the  number  of  days  from  the  standard  date,  and  dividing  the  sum 
of  the  products  by  the  sum  of  the  debts. 

Add  the  average  term  of  credit  to  the  standard  date,  and  the 
result  will  be  the  average  term  of  payment. 

Instead  of  the  earliest  date,  the  first  of  the  month  may  be  used. 


294  AVERAGE   OF   PAYMENTS. 

2.  What  is  the  average  time  at  which  the  following  bills 
become  due:  Feb.  1,  1877,  $200  on  1  mo.  credit;  March  10, 
1877,  $500  on  3  mo.  credit;  April  12,  1877,  $275  on  2  mo. 
credit;  and  May  1,  1877,  $400  on  4  mo.  credit? 

3.  A  merchant  owes  bills  dated  as  follows:  Jan.  1,  1877, 
$500  due  in  2  mo.;    Jan.    15,   1877,   $850  due  in   3  mo.; 
Feb.  20,  1877,  $375  due  in  3  mo.;    and,   Feb.   28,   1877, 
$650  due  in  4  mo.     What  will  be  the  average  time  of  pay- 
ment? 

4.  A  merchant  purchased  goods  of  Cragin  Bros.  &  Co.  as 
follows:  Sept.  10,  1876,  $300  on  4  mo.  credit;  Oct.  15,  1876, 
$400  on  6  mo.  credit;  Nov.  1,  1876,  $750  on  2  mo.  credit; 
and,  Nov.  15,  1876,  $300  on  1  mo.  credit.     What  was  the 
average  time  of  payment? 

5.  Messrs.  J.  Rorbach  &  Son  bought  goods  from  George 
C.  Buell  &  Co.  as  follows:  Sept.  1,  1876,  $600  on  3  mo. 
credit;  Oct.  3,  1876,  $400  on  4  mo.  credit;  Oct.  20,  1876, 
$250  on  2  mo.  credit;  and,  Nov.  10,  1876,  $375  on  1  mo. 
credit.     What  was  the  average  time  of  payment? 

6.  Stevens  &  Shepard  bought  goods  from  the  Russell  Ir- 
win  Manufacturing  Co.  as  follows:  Dec.   10,  1876,  a  bill  of 
$460  on  4  mo.  credit;  Jan.  5,  1877,,  a  bill  of  $200  on  3  mo. 
credit;   Jan.  30,   1877,  a  bill  of  $200  on  4  mo.  credit;  and, 
Feb.  25,   a  bill  of  $900  on  2  mo.  credit.     What  was  the 
average  time  of  payment? 

7.  Bought  goods  of  Carson,  Pirie  &  Co.  as  follows:  Jan. 
25,   1877,  $850  on  4  mo.  credit;  Feb.  15,  1877,  $600  on  3 
mo.  credit;  March   20,    1877,    $500   on   4  mo.  credit;  and, 
April  10,  1877,  $960  on  2  mo.  credit.     What  was  the  aver- 
age time  of  payment? 

8.  May  1,  1877,  Mr.  S.  purchased  goods  to  the  amount  of 
$2400  on  the  following  terms:  \  payable  in  cash,  \  payable 
in  2  months,  and  the  balance  in  6  months.     When  may  the 
whole  be  equitably  paid  by  one  payment? 


AVERAGE   OF    ACCOUNTS. 


295 


AVERAGE  OF  ACCOUNTS. 

465.    1.  What  should  be  the  date  of  a  note  given  to  settle 
the  following  account? 

Dr.  W.  H.  STEVENS.  Or. 


1877. 

1877. 

May 

5 

To  Mdse. 

50 

00 

May 

15 

By  Cash 

25 

00 

June 

7 

"       "      2  mo. 

140 

00 

June 

10 

"   Draft,  10  da. 

100 

00 

June 

20 

"       "      1    " 

150 

00 

June 

30 

"       " 

100 

00 

PROCESS.     (By  Products.) 


Due. 

Amount. 

Days. 

Product. 

Paid. 

Amount. 

Days. 

Product. 

May 
Aug. 
July 

5 

7 
20 

$  50 
140 
150 

94 
0 
17 

4700 
2550 

May 
June 
June 

3150 

15 

23 
30 

.    ^- 

$  25 
100 
100 

84 
45 
38 

ft 

2100 
4500 
3800 

340 
225 

7250 

225 

10400 
7250 

115 

L5  =  27T5- 

3150 

Aug.  7  +  28  days  =  Sept.  4,  the  average  time. 

ANALYSIS. — From  the  dates  at  which  the  various  amounts  become 
due,  we  select  the  latest,  which  is  Aug.  7,  for  the  assumed  time  of  set- 
tlement, and  multiply  each  amount  by  the  number  of  days  intervening 
between  that  date  and  the  time  when  each  item  of  the  account  becomes 
due.  The  debit  side  of  the  account  shows  there  is  due  $340  and  the 
use  of  $1  for  7250  days,  and  the  credit  side  shows  that  $225  has  been 
paid,  and  that  the  debtor  is  entitled  to  the  use  of  $1  for  10400  days,  if 
the  time  of  settlement  is  Aug.  7.  Subtracting  the  amounts,  there  is 
shown  to  be  $115  due,  and  the  debtor  is  entitled  to  the  use  of  $1  for 
3150  days.  Therefore,  he  should  not  be  required  to  pay  the  account 
until  the  time  when  the  use  of  $115  is  equal  to  the  use  of  $1  for  3150 
days,  which  is  28  days.  28  days  after  Aug.  7  is  Sept.  4. 


I 


296 


AVERAGE   OF   ACCOUNTS. 


RULE. — Multiply  each  amount  due  by  the  number  of  days  in- 
tervening between  the  time  it  becomes  due  and  the  latest  date  at 
which  any  sum  on  either  side  of  the  account  becomes  due. 

Divide  the  difference  between  the  sum  of  the  products  of  the 
debit  and  credit  side  of  the  account,  by  the  balance  due  on  the 
account.  The  quotient  will  be  the  average  term  of  credit. 

1.  When  the  balances  are  both  on  one  side  of  the  account,  the  term 
of  credit  is  to  be  counted  backward  from  the  date  at  which  the  first 
amount  becomes  due,  but  forward  from  that  date  if  the  balances  are 
on  opposite  sides. 

2.  The  average  term  of  credit  may  also  be  found  by  reckoning 
interest  upon  each  sum  due  for  the  number  of  days  intervening  be- 
tween the  time  it  becomes  due,  and  the  earliest  date  at  which  any 
sum  becomes  due;  then  dividing  the  balance  of  the  interest  by  the 
interest  on  the  balance  of  the  account  for  one  day.     This  is  called  the 
Method  by  Interest.     The  result  is  the  same  whether  the  average  term 
of  credit  is  found  by  the  method  by  products  or  by  interest. 

2.  Find  the  average  term  of  credit  of  the  following  account : 
Dr.  OLMSTEAD  &  BISHOP.  Or. 


1877; 

1877. 

Jan. 

5 

To  Mdse.,  2ino. 

$375 

Jan. 

30 

By  Cash 

$200 

Feb. 

15 

u           «           x     u 

200 

Mar. 

15 

"       «« 

600 

Feb. 

25 

4    " 

800 

Apr. 

1 

«       tt 

200 

Mar. 

30 

3    " 

450 

3.  Find  by  both  methods  when  the  balance  of  the  follow- 
ing account  becomes  due. 

Dr.  HAMILTON  &  MATTHEWS.  Or. 


1876. 

1877. 

Nov. 

1 

To  Mdse.,  4  mo. 

$1600 

Jan. 

15 

By  Accept'  ce,  2  mo. 

$2000 

Dec. 

3 

3  " 

3800 

Mar. 

20 

«        2  " 

5000 

1877. 

Jan. 

15 

4  " 

5500 

Mar. 

1 

3  ,* 

1500 

AVERAGE   OF   ACCOUNTS.  297 

4.  When  should  interest  begin  on  the  following  account  ? 
Dr.    JAMES  HOWARD,  in  ace't  with  HIRAM  SIBLEY.     Or. 


1877. 

1877. 

Apr. 

10 

To  Mdse. 

$150 

Apr. 

12 

By  Cash 

$250 

Apr. 

30 

«        « 

400 

May 

1 

"        " 

200 

May 

16 

«        u 

100 

June 

7 

1C                  <l 

400 

June 

24 

" 

500 

5.  Find  the  average  term  of  credit  of  the  following  account? 
Dr.  BREYFOGLE  &  Co.  Or. 


1877. 

1877. 

Jan. 

1 

To  Mdse.,  1  mo. 

$  500 

Feb. 

3 

By  Cash 

8500 

Jan. 

20 

"       3   " 

850 

Feb. 

28 

"       «« 

200 

Feb. 

15 

2    « 

1500 

May 

15 

"  Draft,  1  mo. 

1200 

Apr. 

3 

4   " 

2500 

6.  When  should  interest  begin  on  the  following  account  ? 
Dr.  BURK,  FITZSIMONS,  HONE  &  Co.  Or. 


1877. 

1877. 

Feb. 

1 

To  Mdse. 

$1800 

Feb. 

20 

By  Cash 

$3000 

Mar. 

15 

"        "      1  mo. 

3000 

May. 

18 

"  Accept'ce,  2  mo. 

8000 

Mar. 

20 

«        4     « 

4800 

Apr. 

3 

«      4    " 

6000 

7.  What  will  be  the  cash  balance  of  the  following  account, 
Jan.  1,  1878,  interest  at  Q%  ? 


Dr. 


PRATT  J.  NELSON. 


Or. 


1877. 

1877. 

July 

10 

To  Mdse.,  2  mo. 

$500 

July 

20 

By  Cash 

$  400 

Aug. 

1 

3    " 

700 

Aug. 

20 

(C                 It 

1000 

Sept. 

8 

<«        «        !    « 

800 

Sept. 

20 

2    « 

600 

4:66.  1 .  If  two  men,  who  have  equal  sums  invested  in  the 
same  business,  gain  $100,  what  is  each  man's  share  of  the 
gain? 

2.  If  one  man  furnishes  J-  of  the  capital,  and  another  -|  of 
it,  and  the  gain  is  $1200,  what  should  be  the  gain  of  each? 

3.  Mr.  A.  furnishes  $3000  of  the  capital,  and  Mr.  B.  fur- 
nishes the  balance,  which  is  $5000.     What  part  of  the  profits 
should  each  receive? 

4.  Four  partners  furnish  money  in  the  proportion  of  $2000, 
$3000,  $4000,  and  $5000  respectively.     What  part  of  the 
gain  should  each  one  receive  ? 

5.  Three  men  engage  in  business  and  furnish  the  following 
sums  respectively:    A,  $5000;    B,  $4000;    C,  $3000.     How 
much  of  the  gain  should  each  receive  if  $1200  was  gained 
during  the  year? 

6.  The  cost  of  a  pasture  was  $27.     A  had  in  it  5  cows  for 
3  weeks,  and  B  3  cows  for  4  weeks.     What  should  each  one 
pay? 

7.  The  profits  of  a  company  were  $800  for  a  certain  time. 
What  share  of  the  profits  did  each  partner  receive,  if  the 
capital  contributed  by  them  was  $900,  $700,  and  $800  re- 
spectively ? 

8.  A  and  B  formed  a  partnership  after  A  had  been  doing 
business  alone  for  6  months.     A  had  $5000  invested  during 
the  year,  and  B  had  $10000  invested  for  6  months.     The 
gain  was  $5000.     What  was  each  one's  share  ? 

(298) 


PARTNERSHIP.  299 


DEFINITIONS. 

467.  A  Partnership  is  an  association  of  two  or  more 
persons,  for  the  purpose  of  conducting  business. 

468.  Partners  are  the  persons  associated  in  business. 
They  are  called  collectively  a  company,  a  firm  or  house. 

469.  The  Capital  is  the  money  employed  in  business. 

* 

470.  PRINCIPLE. — Partners  share  the  gains  and  losses  in  pro- 
portion to  the  amount  of  the  capital  each  invests,  and  the  length 
of  time  it  is  employed. 

CASE  I. 

471.  When  the  capital  of*  each  partner  is  employed 
tor  the  same  time. 


WRITTEN    EXERCISES. 

1.  A,  B  and  C  are  partners,  having  furnished  $5000, 
$6000  and  $8000  capital  respectively.  If  during  the  year 
they  gain  $2850,  what  is  each  partner's  share  of  the  gain? 

PROCESS.  ANALYSIS.— 

$5000  +  $6000  +  $8000  =  $19000  ® ™e  Th  o^l  d 

iW<ro   or  T9   of  $2850=   $750,  A's  share.        share  the  gain 

6000     or     6     Of  $2850=    $900,  B's  share.          in  proportion  to 

the  amount  of 
A0o°o°o   or  A  of  $2850  =  $1200,  C's  share.        capital  he  con. 

$2850,  Whole  gain.    tributed,wefind 
what  part  of  the 

whole  capital  each  partner  contributed.  A  furnished  T5^  of  the  capital, 
and  is  therefore  entitled  to  T5g  of  the  gain,  or  $750.  B  furnished  T6¥ 
of  the  capital,  and  is  entitled  to  r%  of  the  gain,  or  $900.  C  furnished 
T8y  of  the  capital,  and  is  therefore  entitled  to  1%  of  the  gain,  or  $1200. 


300  PARTNERSHIP. 

KULE. — Find  such  a  part  of  the  gain  or  loss  as  each  partner's 
capital  is  of  the  ivhole  capital. 

The  result  will  be  each  partner's  gain  or  loss. 

2.  A,  B  and  C   engaged  in  business,  employing  $20000 
capital,  of  which  A  furnished  $7000,  B  $7000,  and  C  $6000. 
They  gained  in  one  year  $6000.     What  was  each  partner's 
share? 

3.  Three  men  engage  in  business.     A  furnishes  $3000  of 
the  capital,  B  $6000  and  C  $4000.     If  they  gain  $2600,  what 
is  each  partner's  share? 

4.  Three  men  engaged  in  land  speculation.     A  furnished 
$10000,  B  $8000  and  C  $12000.     They  lost  in  one   year 
$6000.     What  was  the  loss  of  each  partner? 

5.  A,   B  and  C  furnish  capital  to  engage  in  business  as 
follows:   A  $2500,  B  $2000  and  C  $3500.     If  the  firm  loses 
$640, what  is  the  loss  of  each  partner? 

6.  A,  B,  C  and  D  engaged  in  buying  produce.     A  con- 
tributed $8000,  B  $10000,  C  $9000  and  D  $13000.     They 
gained  $3000.     What  was  each  partner's  share  of  the  gain? 

7.  D  and  G  furnish  capital  to  engage  in  business  and  L 
does  the  work  for  ^  of  the  profits;  D  contributes  $8000  and 
G  10000  of  the   capital.      They  gain  $5400.    What  is  each 
one's  share  of  the  gain  ? 

8.  E,  F  and  G  bought  a  block  of  stores  for  $46000.     E 
furnished  f  of  the  money,  F  $11500  and  G  the  rest.     The 
property  was  sold  for  $48300.     What  was  the  gain  of  each? 

9.  A,  B  and  C  engage   in  business.     A  furnishes   $6470, 
B  $5420  and  C  $3410  capital.     If  they  gain  $6490.75,  what 
is  the  gain  of  each  ? 

10.  Four  persons  rented  conjointly  a   pasture  containing 
125  A.  60  sq.  rd.,  for  $3.75  an  acre.     A  fed  125  sheep  upon 
it,  B  145  sheep,  C  175  sheep,  and  D  340  sheep.     How  much 
rent  should  each  one  pay? 


PAKTNEESHIP.  301 

11.  Three  men  engaged  in  business.     A  furnished  $6000 
and  B  $8000.     They  gained  $4200,  of  which  C's  share  was 
$1400.     What  was  the  gain  of  A  and  B  and  C's  stock? 

12.  Five   men   trade   in   partnership.     A  furnishes  $500, 
B  $600,  C  $800,  D  $1000  and  E  $1200  capital.     They  gain 
$2750.     What  is  the  gain  of  each  partner  ? 

13.  A,  B  and  C  bought  a  farm  in  partnership.     A  paid  \ 
the  purchase  money,  B  ^  and  C  the  rest.     They  sold  it  at  a 
gain  of  $3000.     What  was  each  one's  share  of  the  profit  ? 

CASE  II. 

472.  When  the  capital  of  the  partners  is  employed 
for  different  periods  of  time. 

WRITTEN    EXERCISES. 

1.  A  began  business  with  $6000  capital.  At  the  end  of 
6  months  he  took  in  B  as  a  partner,  who  furnished  $5000 
additional  capital.  If  the  gain,  after  6  months  more,  was 
$3400,  what  was  each  partner's  share  of  the  gain? 

PROCESS.  ANALYSIS. — A's  cap- 

ital of  $6000  was  used 

*6000X12  =  *72000  for  12  months,  and  was 

$5 00 OX     6  =  $30000  therefore  equal  to  the 

$102000  use  of  $72000  for   1 

month.      B's    capital 

of  $  3  4  0  0  =  $  2  4  0  0,  A's  share.      of  $500o  was  used  for 
of  $ 3 40 0  =  $  1 0 0 0,  B's.  share.       6  months,  which  was 

equal   to   the    use   of 

$30000  for  1  month.  Both  together  had  invested  sums  of  money 
which  were  equal  to  the  use  of  $102000  for  1  month,  of  which  A  con- 
tributed a  sum  equal  to  $72000  for  1  month,  or  •£?%,  and  he  was  there- 
fore entitled  to  ^  of  the  gain,  or  $2400.  B ,  contributed  a  sum 
equal  to  $30000  for  1  month,  or  ^V,  and  was  therefore  entitled  to 
T3o\  of  the  gain,  or  $1000. 


302  PARTNERSHIP. 

RULE. — Find  such  a  part  of  the  entire  gain  or  loss,  for  each 
partner's  share  of  the  gain  or  loss,  as  the  capital  of  each  partner 
for  a  unit  of  time,  is  of  the  entire  capital  for  a  unit  of  time. 

2.  A  engaged  in  business  with  a  capital  of  $4000.     After 
3  months  he  took  in  B  with  a  capital  of  $6000,  and  in  6 
more,  C  became  a  partner,  with  a  capital  of  $8000.     At  the 
end  of  18  months  the  profits  were  $9360.     What  was  each 
partner's  share  of  the  gain  ? 

3.  A,  B  and  C  engage   in  business  together.     A  puts  in 
$4000  capital  for  8  months,  B  $6000  for  7  months,  and  D 
$3500  for  one  year.     If  they  gain  $2320,  what  is  each  part- 
ner's share  of  the  gain? 

4.  B,   C   and   D  entered    into    partnership,    furnishing  a 
joint  capital  of  $5875,  of  which  B  furnished  20%,  C  35%, 
and  D  the  rest.     B's  capital  was  employed  15  months,  C's 
9  months,  and  D's  10  months.     They  lost  $2502.75.     What 
was  each  partner's  loss? 

5.  A,  B  and  C  took  a  contract  to  build  a  block  of  stores. 
A  furnished  20  men  for  3  months,  B  25  men  for  3J  months, 
and  C  15   men  for  4  months.     After  paying  the  expenses 
the  profits  were  $1475.     What  was  the  share  of  each? 

6.  A,  B  and  C   lost   $8500   by  speculating  in  real  estate. 
A  furnished  $5000  of  the  capital  which  was  employed  for  1 
year,  B  $8000  for  10  months,  and  C  $10000  for  6  months. 
What  was  each  one's  share  of  the  loss? 

7.  A,  B  and  C  engaged  in  manufacturing  rope  and  cord- 
age.    A  invested  $4500  for  6  months,  B  $5000  for  8  months, 
and  C  $6500  for  7  months.     They  gained  $4500.     What  was 
the  gain  of  each  partper? 

8.  G,  L  and  F  entered   into   partnership.      G  furnished 
$1200,  L  $1500,  and  F  $3000.     After  6  months  F  withdrew 
$2000  of  his  capital.     If  at  the  end  of  a  year  the  profits  were 
$2200,  what  part  of  the  profits  belonged  to  each  partner  ? 


01 


473.  1.  A  was  employed  on  a  piece  of  work  6  days,  and 
B  12  days  on  the  same  work.  How  does  the  number  of 
days  A  worked  compare  with  the  number  of  days  B  wras 
employed? 

2.  A  laborer  earned  $12  a  week,  and  spent  $6.     How 
does  what  he  spent  compare  with  what  he  earned? 

3.  How  does  $3  compare  with  $9?     $4  with  $12?     $6 
with  $18? 

4.  How  does  2  compare  with  10?     3  with  18?     5  with 
25? 

5.  What  relation  has  2  to  12?     3  to  21?     4  to  28? 

6.  What  is  the  relation  of  3  to  24?     6  to  30?     7  to  35? 

7.  How  does  8  compare  with  2?     What  is  the  relation 
of  8  to  2? 

8.  How  does  9  compare  with  3?     What  relation  has  9 


to  3? 

9. 

10. 

11. 


What  relation  has  24  to  8?     30  to  6?     25  to  4? 
What  is  the  relation  between  5  and  7  ?     Ans.  -f-  or  |-. 
What  is  the  relation  between   6   and   9?     Between  8 
and  9? 

12.  What  is  the  relation  of    8  to  9?     Between    8  and  9? 

13.  What  is  the  relation  of  12  to  4?     Between  12  and  4? 
L4.  What  js  the  relation  of  15  to  5?     Between  15  and  5? 

15.  What  is  the  relation  of  16  to  8?     Between  16  and  8? 

16.  What  is  the  relation  of  25  to  5?     Between  25  and  5? 

(303) 


304  RATIO. 


DEFINITIONS. 

474.  Ratio  is  the  relation  of  one  number  to  another  of 
the  same  kind. 

1.  This  relation  is  expressed  either  as   quotient  of    one   number 
divided  by  the  other,  and  is  called  Geometrical  Ratio,  or  simply  Ratio, 
or,  as  the  difference  between  two  numbers,  and  is  called  Arithmetical 
Ratio. 

2.  When  it  is  required  to  determine  what  the  relation  of  one  number  to 
another  is,  it  is  evident  that  the  first  is  the  dividend,  and  the  second  the 
divisor. 

3.  When  it  is  required  to  determine  the  relation  between  two  numbers, 
either  may  be  regarded  as  dividend  or  divisor. 

4.  The  first  number  is  commonly  regarded  as  the  dividend. 

475.  The  Terms  of  a  Ratio  are  the  numbers  com- 
pared. 

476.  The  Antecedent  is  the  first  term. 

Thus,  in  "  What  is  the  ratio  of  6  to  8  ?  "  6  is  the  antecedent. 

477.  The  Consequent  is  the  second  term. 

Thus,  in  "  What  is  the  ratio  of  6  to  8  ?  "  8  is  the  consequent. 

478.  The  Sign  of  ratio  is  a  colon  (:). 
Thus,  the  ratio  of  12  to  6  is  expressed,  12  :  6. 

The  colon  ( : )  is  sometimes  regarded  as  the  sign  of  division  without 
the  line.     Thus,  12  :  8  is  regarded  as  12  -5-  8. 

479.  The    antecedent    and    consequent    together    form    a 
Couplet. 

480.  PRINCIPLES. — 1.    The  terms  of  a  ratio  must  be  like 
numbers. 

2.  The  ratio  is  an  abstract  number. 

3.  Multiplying  or  dividing  both  terms  of  a  ratio  by  the  same 
number  does  not  change  the  ratio  of  the  numbers. 


BATIO.  305 


EXERCISES. 

481.    1.  What  is  the  ratio  of  3  to  6?     5  to  10?     7  to  21  ? 

2.  What  is  the  ratio  of  $3  to  $10?     12  Ib.  to  6  lb.?     27 
bush,  to  9  bush.  ? 

3.  What  is  the  ratio  of  7  to  35?     24  to  48?     13  to  39  ? 

4.  If  the  antecedent  be  20,  and  the  consequent  15,  what  is 
the  ratio? 

5.  What  is  the  ratio  when  the  antecedent  is  45,  and  the 
consequent  25? 

6.  What  is  the  ratio  of  f  to  f  ?     f  to  f  ?     f  to  T7T? 

Fractions  should  be  reduced  to  similar  fractions.     They  will  then 
have  the  ratio  of  their  numerators. 

7.  What  is  the  ratio  of  5£  to  3|?     7-J-  to  61?     9|  to5i? 

8.  What  ratio  will  the  work  of  12  men  sustain  to  that  of 
8  men? 

9.  What  will  be  the  ratio  of  8  yd.  to  24  yd.  ?     6f  yd.  to 
9yd.? 

10.  When  the  antecedent  is  3  and  the  ratio  -f ,  what  Is  the 
consequent? 

11.  When  the  consequent  is  8  and  the  ratio  -f-,  what  is  the 
antecedent  ? 

12.  When  the  antecedent  is  %  and  the  ratio  ^,  what  is  the 
consequent? 

13.  What  number  has  to  3  the  ratio  of  5  to  6? 

14.  What  number  has  to  5  the  ratio  of  4  to  12? 

15.  What  number  has  the  ratio  to  12  that  3  has  to  1? 

16.  If  two  numbers  have  the  relation  of  6  to  8,  and  the 
first  is  12,  what  is  the  other? 

17.  What  number  has  to  12  the  ratio  of  8  to  9? 

18.  If  two  numbers  have  the  relation  of  10  to  15,  and  the 
antecedent  is  40,  what  is  the  consequent? 

20 


' 


d:OTsItJ:WTi17 


482.    1.  What  two  numbers  have  the  same  relation  to  each 
other  as  3  to  6  ?     As  2  to  8  ?     As  7  to  21  ? 

2.  What  two  numbers  have  the  same  ratio  as  5  to  15? 
6  to  30?     12  to  48?     12£to25?     2£  to  4J?     12^  to  50? 

3.  What  number  has  the  same  relation  to  6  that  3  has 
to  9? 

4.  What  number  has  the  same  relation  to   5  that  7  has 
to  14? 

5.  What  number  has  the  same  relation  to  -f  that  4  has 
to  8? 

6.  To  what  number  has  5  the  same  relation  that  3  has 
to  9? 

7.  To  what  number  has  2^  the  same  relation  that  7  has 
to  21? 

8.  24  is  to  7  as  12  is  to  what  number? 

9.  12  is  to  5  as  what  number  is  to  15? 

10.  If  the  cost  of  9  yards  of  cloth  is  $5,  how  will  the  cost 
of  18  yards  compare  with  that  sum? 

11.  If  10  men  can  earn  $30  per  day,  what  ratio  will  the 
earnings  of  15  men  bear  to  that  sum? 

12.  Write  two   equal  ratios;    multiply  the   first   and  last 
terms  together ;  multiply  the  second  and  third  terms  together. 
How  do  the  products  compare? 

13.  Write    two  other  equal  ratios;    multiply  as   before. 
How  do  the  products  compare? 

(306) 


PROPORTION.  307 


DEFINITIONS. 

483.  A  Proportion  is  an  equality  of  ratios. 

Thus,  9  :  18  ==  6  : 12  is  a  proportion. 

484.  The  Sign  of  proportion  is  a  double  colon  (  :  :  ). 
The  double  colon  ( :  : )  may  be  regarded  as  the  extremities  of  the 

sign  of  equality  (  =  ).     It  is  written  between  the  ratios. 

A  proportion  must  have  four  terms,  viz:  two  antecedents, 
and  two  consequents. 

Any  four  numbers  that  can  be  formed  into  a  proportion 
are  called  proportionals. 

485.  The  Antecedents  of  a  proportion  are  the  antece- 
dents of  the  ratios,  or  the  first  and  third  terms. 

Thus,  in  the  proportion  5  :  10  :  :  7  :  14,  5  and  7  are  the  antecedents. 

486.  The  Consequents  of  a  proportion  are  the  conse- 
quents of  the  ratios,  or  the  second  and  fourth  terms. 

Thus,  in  the  proportion  5  :  10  :  :  7  :  14,  the  consequents  are  10  and  14. 

487.  The  Extremes  of  a  proportion  are  the  first  and 
fourth  terms. 

Thus,  in  the  proportion  7  :  8  :  :  14  :  16,  7  and  16  are  the  extremes. 

488.  The  Means  of  a  proportion  are  the  second  and 
third  terms. 

Thus,  in  the  proportion  7  :  8  :  :  14  :  16,  8  and  14  are  the  means. 

489.  PRINCIPLES. — 1.    The  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

2.  The  product  of  the  extremes  divided  by  either  mean  gives 
the  other  mean. 

3.  The  product  of  the  means  divided  by  either  extreme  gives  the 
other  extreme. 


308 


PBOPORTION. 


EXERCISES. 


Find  the  term  that  is  wanting  in  the  following: 


1. 

18 

:  24  :: 

6  : 

C  )• 

9. 

(  ) 

:  14  :: 

16  :  35. 

2. 

9  : 

18::  I 

\  / 

10. 

V.  / 

14 

:  23  :  : 

C  )  :  69. 

3. 

8  : 

18  :: 

[  ). 

11. 

13 

:  (  )  :: 

15  :  65. 

4. 

/  \ 

•  18  • 

•  7 

15 

12. 

i  . 

5  ::  (  " 

:  6. 

5. 

1  : 

8  : 

24. 

13. 

f  : 

•  v  > 

:  10. 

6. 

15 

:  18  :: 

(  ) 

:  16. 

14. 

1  = 

f::(J 

:  15. 

7. 

17 

:  19  :: 

15 

:  ()• 

15. 

6  : 

|  ::  5  : 

(  )• 

8. 

25 

=  (  )  : 

:  16 

:  25. 

16. 

13 

:7::( 

):8. 

17. 

5  men  :  7  men  :  : 

8*: 

(  )• 

18.  $21.16  :  $15.20  ::  f  :  (  ). 

19.  80.51  A.  :  21.15  A.  ::  (  ) 

20.  16  Ib.  3  oz.  :  18  Ib.  2  oz.  : 

21.  5  gal.  3  qt.  :  (  )  : :  5  :  9. 

22.  14  f0  :  (  )  : :  6  :  15. 


:  2. 

() 


7. 


SIMPLE  PROPORTION. 

490.  A  Simple  Ratio  is  a  ratio  between  any  two 
numbers. 

Thus,  6: 8,  $10: $8,  5  Ib.  6  oz.:  7  Ib.  3  oz.,  are  simple  ratios. 

491.  A  /Sim%)le  Proportion  is  an  equality  between 
two  simple  ratios. 

492.  A  Direct  Proportion  is  one  in  which  each 
term  increases  or  diminishes,  as  the  one  on  which  it  depends 
increases  or  diminishes. 

Thus,  proportions  involving  quantity  and  cost,  men  and  work  done, 
etc.,  are  direct  proportions,  for  as  the  quantity  increases  or  diminishes, 
the  cost  increases  or  diminishes,  and  as  the  number  of  men  increases  or 
diminishes,  the  amount  of  work  done  will  increase  or  diminish. 


PKOPORTION. 


309 


493.  An  Inverse  Proportion  is  one  in  which  each 
term  increases  as  the  term  upon  which  it  depends  diminishes, 
or  diminishes  as  it  increases. 

Thus,  in  the  problem,  "If  6  men  can  mow  a  field  of  grass  in  9 
days,  how  long  will  it  take  9  men  to  mow  it,"  as  the  number  of  men 
increases,  the  number  of  days  required  to  do  the  work  decreases,  and  the 
proportion  is  an  inverse  proportion. 


WRITTEN   EXERCISES. 


494.    1.  If  8  yd.  of  silk  cost  $24,  what  will  15  yd.  cost? 


PROCESS. 


yd.    yd. 

(1)  8  :  15  : 

yd.    yd. 

(2)  15  :  8  : 


24  :  (  ) 


(  )  :  24 


The  term  wanting  (1) 
The  term  wanting  (2) 


=  $45 


ANALYSIS. — It  is  evi- 
dent that  8  yd.  have  the 
same  relation  to  15  yd. 
that  the  cost  of  8  yd.  has 
to  the  cost  of  15yd.  Hence 
we  have  the  proportion, 
8  yd.  :  15  yd.  ::  $24,  the 
cost  of  8  yd.  :  the  cost  of 
15yd.,  or  15  yd.  :  8yd.  :: 
the  cost  of  15  yd.  :  $24,  the  cost  of  8  yd.  To  find  the  cost  of  15  yards, 
the  term  wanting,  we  divide  the  product  of  the  means  by  the  extreme, 
as  in  (1) ;  or  the  product  of  the  extremes  by  the  mean,  as  in  (2). 

2.  If  5  men  can  cut  a  quantity  of  wood  in  18  days,  in  how 
many  days  could  12  men  do  the  same  work? 

ANALYSIS.  —  It  is  evident 
that  exactly  in  proportion  as 
the  number  of  men  is  increased, 
the  number  of  days  required  to 
do  the  work  is  diminished,  and 
therefore  5  men  :  12  men  ::  the 


PROCESS. 

men.  men.     days.  days. 

(1)  5  :  12  ::  (  )  :  18 

men.  men.  days.  days. 

(2)  12  :  5  ::  18  :  (  ) 


Term  wanting  =  -Lf4p.  —  7-J-  da.      days  it  will  require  12  men  to 

do  the  work  :  18  the  number  of 
days  required  for  5  men  to  do  the  work.     Or, 

12  men  :  5  men  : :  18  days,  the  number  of  days  it  requires  5  men  to 
do  the  work  :  the  number  of  days  12  men  require  to  do  the  work. 


310  PKOPORTION. 

RULE. — Express  the  ratio  between  the  two  numbers  that  are  like 
numbers.  Consider,  from  the  conditions  of  the  problem,  whether 
the  proportion  is  direct  or  indirect,  and  arrange  the  other  number 
and  the  term  wanted  so  that  the  two  ratios  will  be  equal. 

Divide  the  product  of  the  extremes  or  means  by  the  single  ex- 
treme or  mean.  The  result  will  be  the  term  wanted. 

Problems  in  proportion  are  sometimes  regarded  as  illustrations  of 
cause  and  effect,  in  which  two  causes  and  their  corresponding  effects 
are  compared,  giving  the  following  proportion: 

1st  cause  :  2d  cause  : :  1st  effect :  2d  effect. 

3.  If  6  men  earn  $75  in  one  week,  how  much  will  10  men 
earn  in  the  same  time? 

4.  If  16  yards  of  cloth  cost  $20,  what  will  be  the  cost  of  7 
yards  ? 

5.  A  man  can  buy  45  sheep  for  $112.50.     How  much  will 
18  sheep  cost  at  the  same  rate? 

6.  If  8  horses  consume  15  tons  of  hay  in  6  months,  how 
much  hay  will  14  horses  consume  in  the  same  time? 

7.  If  6  men  can  do  a  piece  of  work  in  45  days,  how  many 
days  will  it  take  11  men  to  do  the  same  work? 

8.  If  10  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  13  men  do  the  same  work? 

9.  How  many  men  will  it  require  to  build  60  rods  of  wall 
in  the  same  time  that  8  men  can  build  40  rods? 

10.  If  8  men  can  dig  a  ditch  in  15  days,  how  many  days 
will  it  take  13  men  to  dig  it? 

11.  If  6  bushels  of  wheat  can  be  bought  for  $7.32,  how 
many  bushels  can  be  bought  for  $45  ? 

12.  How  many  barrels  of  apples  can  be  bought  for  $2250, 
if  15  barrels  cost  $33.75? 

13.  If  it  requires  13  men  to  lay  a  certain  number  of  bricks 
in  28  days,  how  many  days  will  it  take  9  men  to  lay  the  same 
number  ? 


COMPOUND    PROPORTION.  311 


14.  If  165  bushels  of  potatoes  can  be  raised  on  1^  acres 
of  ground,  how  many  bushels  can  be  raised  on  3^  acres  ? 

15.  If  it  requires  1^  acres  of  ground  to  raise  405  bu.  of 
carrots,  how  many  acres  will  it  require  to  raise  975  bu.  ? 

16.  Five  horses   cost  a   man    $626.25.     What  would  be 
the  cost  of  13  horses  at  the  same  rate? 

17.  It  required  26   men  to  build  an  embankment  in  80 
days.     How  long  would  it  require  32  men  to  do  the  same 
work  ? 

18.  It  took   9  horses  to  move  a  stick  of  timber  weighing 
12590  pounds.      How  many  pounds  would   a   stick  weigh 
which  could  be  moved  by  7  horses? 

19.  If  an  ocean  steamer  sails  1775  miles  in  5  days,  how 
many  miles  will  she  sail  in  6^  days? 

20.  If  a  locomotive  runs  96f  miles  in  3|-  hours,  how  many 
miles  will  it  run  in  5£  hours? 

21.  A  dog  is  chasing  a  rabbit,  which  has  45  rods  the  start 
of  the  dog.     The  dog  runs  19  rods  while  the  rabbit  runs  17. 
How  far  must  the  dog  run  before  he  catches  the  rabbit? 

22.  A  cistern  has   3  pipes.     The  first  will  fill  it  in  12 
hours,  the  second  in   16  hours,  and  the  third  in  18  hours. 
If  all  run  together  how  long  will  it  take  them  to  fill  it  ? 

23.  If  it  requires  15  compositors  15  days  to  set  up  a  book 
of  675  pages,  how  many  days  will  they  need  to  set  up  a  book 
of  900  pages? 

COMPOUND  PROPORTION. 

495.  A  Compound  Hatio  is  the  product  of  two  or 
more  simple  ratios. 

496.  A  Compound  Proportion  is  a  proportion  in 
which  either  ratio  is  compound. 

497.  PRINCIPLE.  —  The  product  of  two  or  more  proportions  .is 
a  proportion. 


312  PROPORTION. 


WRITTEN    EXERCISES. 

498.  1.  If  6  men  can  mow  24  acres  of  grass  in  2  days, 
by  working  10  hours  per  day,  how  many  days  will  it  take  7 
men  to  mow  56  acres,  by  working  12  hours  per  day?- 

PROCESS.  ANALYSIS. — A  simple  pro- 

(1)  7:    6::  2    :  (1-f-  days.)        portion  is  a  proportion  that 

(2)  24  :  56  : :  1*  :  (4    days.)        has  but  one  condition-   A  com' 
)ov  10    -in       A       )oi    ,1  pound    proportion    has   more 

(3)  12  : 10  : :  4    :  (3J  days.)        than  one  condition.    The  con. 

Qr  ditions  are  introduced  one  at 

s    n     n    ^  a  time,  therefore  examples  in 

compound  proportion  may  be 
(Q)     <  Z4.0O  >  ..Z.Og-  solved  as  several  simple  pro- 

I  19*10  I 

portions.     The  first  condition 

Means      6  X  56  X  10  X  2  in  this  examPle  is :  ,If  6  men 

=  3^-  can  mow  24  acres  of  grass  in 

Extremes  7  X  24  X  12  2  days,  how  long  will  it  take 

7  men  to  do  the  same  work? 

This,  solved  by  simple  proportion,  (1),  gives  If  days.  The  second 
condition  is:  If  the  men  can  mow  24  acres  of  grass  in  If  days,  how 
long  will  it  take  them  to  mow  56  acres?  This,  solved  by  simple  pro- 
portion, (2),  gives  4  days.  The  third  condition  is:  If  the  work  can 
be  done  by  the  men  in  4  days,  by  working  10  hours  per  day,  how 
many  days  will  it  take  to  do  the  work  if  they  work  12  hours  per  day? 
This,  solved  by  simple  proportion,  (3),  gives  3J  days,  the  time  it  will 
take  7  men  to  mow  56  acres  of  grass,  by  working  12  hours  per  day,  if 
six  men  can  mow  24  acres  in  2  days  by  wording  10  hours  per  day.  Or, 

Since  every  simple  proportion  is  an  equality  of  ratios,  the  product 
of  the  three  proportions,  (1),  (2),  (3), 'will  be  an  equality  of  ratios; 
and,  since  If  and  4  appear  in  both  antecedent  and  consequent,  they 
may  be  omitted,  and  the  simple  proportions  will  assume  the  form  of 
the  compound  proportion,  (4). 

The  problem  may  be  stated,  as  in  the  second  part  of  the  process,  by 
writing  for  the  third  term  the  term  that  is  like  the  one  sought,  and  by 
arranging  the  others  in  couplets,  considering  their  relation  to  the  ratio 
between  the  third  term  and  the  term  sought. 


COMPOUND   PROPORTION.  313 

RULE. — Solve  by  successive  simple  proportions,  introducing  the 
conditions  one  at  a  time.  Or, 

Use  for  the  third  term  the  number  which  is  of  the  same  kind 
as  the  term  required. 

Arrange  the  like  numbers  in  couplets,  as  in  simple  proportion. 

The  product  of  the  means  divided  by  the  product  of  the  ex- 
tremes will  be  the  term  required. 

Problems  in  compound  proportion  are  readily  solved  by  cause  and 
effect.  Example  1,  stated  by  cause  and  effect,  is  as  follows  : 

Is*  Cause.        2d  Cause.  1st  Effect.          2d  Effect. 

6  men    ^        7  men    ^          C  C 

2  days     >•  j    ( )  days     >•  :  :  •<  24  acres  :  X  56  acres 

10  hours)      12  hours }          (_  (_ 

Means      6  X  2  X  10  X  56  _ 
Extremes      7  X  12  X  24 

2.  If  15  men  can  dig  a  ditch  in  45  days  by  working  10 
hours  a  day,  how  many  days  will  it  take  20  men  to  dig  it, 
by  working  12  hours  a  day? 

3.  If  a  block  of  granite  6  feet  long,  3  feet  wide,  and  2  feet 
thick,  weighs  5940  pounds,  what  will  be  the  weight  of  a 
block  of  the  same  kind,  which  is  9  feet  long,  4  feet  wide 
and  3  feet  thick? 

4.  If  I  place  $1500  at  interest  for  18  months  and  receive 
$135  interest,  what  sum  must  I  place  at  interest  at  the  same 
rate,  so  that  I  may  receive  $275  interest  in  8  months  ? 

5.  If  it    cost   $180   to  support   5    grown  persons    and   3 
children  for  3  weeks,  what  will  it  cost  to  support  8  grown 
persons  and  6  children  for  7  weeks,  allowing  that  it  costs  ^ 
as  much  to  support  a  child  as  a  grown  person? 

6.  If  20  men  working  8  hours  a  day,  can  dig  a  trench 
65  feet  long,  9  feet  wide  and  6  feet  deep,  in  25  days,  how 
many  days  will  it  take  25  men,  working  10  hours  a  day,  to 
dig  a  trench  75  feet  long,  8  feet  wide,  and  7  feet  deep? 


814  PBOPORTION. 

7.  If  it  costs  $240  to  board  16   persons   5  weeks,  how 
much  will  it  cost  to  board  9  persons  22  weeks? 

8.  If  $800  placed  at   interest,   amounts   to  $880  in   15 
months,  what  sum  must  be  placed  at  interest  at  the  same 
rate,  to  amount  to  $975  in  one  year? 

9.  If  it  requires  275  yards  of  cloth  f  yd.  wide  to  make 
75  garments,  how  many  yards  of  cloth  1J  yd.  wide,  will  it 
require  to  make  215  such  garments? 

10.  If  a  bin  which  is  8  feet  long,  6  feet  wide  and  8  feet 
deep,  holds  309  bushels  of  wheat,  how  many  bushels  will  a 
bin  hold  that  is  14  feet  long,  8  feet  wide  and  9  feet  deep? 

11.  If  15  men,  working  10  hours  a  day,  can  do  a  certain 
piece  of  work  in  18  days,  how  many  days  will  it  require  for 
13  men  to  do  the  same  work,  by  working  8  hours  a  day? 

12.  If  12  horses  consume  40  bushels  of  oats  in  8  days, 
how  long  will  140  bushels  of  oats  last  16  horses? 

13.  If  a  regiment  of  1025  soldiers  consume  11500  pounds 
of  bread  in   15    days,  how  many  pounds  will   3   regiments 
of  the  same  size,  consume  in  12  days  ? 

14.  If  the  water  that  fills  a  vat,  which  is  8  feet  long, 
4  feet  wide  and  5  feet  deep,  weighs  10000  pounds,  what  will 
be  the  weight  of  the  water  required  to  fill  a  vat,  which  is 
10  feet  long,  5  feet  wide  and  6  feet  deep  ? 

15.  If  5  horses  eat  as  much  as  6  cattle,  and  8  horses  and 
12  cattle  eat  12  tons  of  hay  in  40  days,  how  much  hay  will 
be  needed  to  keep  7  horses  and  15  cattle  65  days? 

16.  If  15  men  working  6  hours  a  day,  can  dig  a  cellar 
80  feet  long,  60  feet  wide  and  10  feet  deep  in  25  days,  how 
many  days  will  it  require  25  men  working  8  hours  a  day,  to 
dig  a  cellar  120  feet  long,  70  feet  wide  and  8  feet  deep? 

17.  If  52  men  can  dig  a  trench  355  feet  long,  60  feet 
wide  and  8  feet  deep  in  15  days,  how  long  will  a  trench  be, 
that  is  45  feet  wide  and  10  feet  deep,  which  45  men  can  dig 
in  25  days? 


499.  1.  Of  what  number  are  3  and  3  the   factors?    4 
and  4? 

2.  Of  what  number  are  3  and  3  and  3  the  factors?    4  and 
4  and  4? 

3.  What  is  the  product  when  5  is  used  twice  as  a  factor? 

4.  What  is  the  product  or  power,  when  6  is  used  twice  as 
a  factor?     When  8  is  used  twice  as  a  factor? 

5.  What  is  the  product  of  f  X  f  ?     Of  f  X  f? 

6.  What  is  the  product  when  f  is  used  twice  as  a  factor  ? 
When  |  is  used  three  times  as  a  factor? 

7.  What  is  the  product  of  two  4's,  or  the  second  power 
of  4?     What  is  the  product  of  three  5's,  or  the  third  power 
of  5  ?     What  is  the  third  power  of  6  ? 

8.  What  is  the  second  power  of  f  ?     Of  f  ?     Of  f  ? 

DEFINITIONS. 

500.  A  Power  of  a  number  is  the  product  arising  from 
using  the  number  a  certain  number  of  times  as  a  factor, 

501.  The  powers  of  a  number  are  named  from  the  number 
of  times  the  number  is  used  as  a  factor. 

Thus,  when  2  is  used  as  a  factor  twice,  the  product,  4,  is  called  the 
second  power  of  2.  9  is  the  second  power  of  3.  27  is  the  third  power 
of  3. 

The  number  itself  is  called  the  first  power. 

(315) 


316  INVOLUTION. 

502.  The  number  of  times  a  number  is  used  as  a  factor  is 
indicated   by  a   small  figure   called  an  Exponent,  written  a 
little  above  and  at  the  right  of  the  number. 

Thus,  32  means  the  second  power  of  3;  54,  the  fourth  power  of  5,  etc. 

Inasmuch  as  the  area  of  a  square  is  the  product  of  two  equal  factors, 
and  the  volume  of  a  cube  is  the  product  of  three  equal  factors,  the  second 
power  of  a  number  is  also  called  the  square,  and  the  third  power  the 
cube  of  the  number. 

503.  Involution  is  the  process  of  finding  the  power  of 
a  number. 

WRITTEN    EXERCISES. 

504.  1.  Find  the  third  power  of  15. 

PROCESS.  ANALYSIS. — To    find    the    third 

icv/iK\/iK Q  Q  7  K       power  of  a  number  is  to  find  the 

1O/\1O/\1O  —  O  O  i  u  . 

product,  when  the  number  is  used  3 

times  as  a  factor.     Therefore,  the  third  power  of  15  will  be  15  X  15 
X  15,  which  is  equal  to  3375. 

2.  Find  the  third  power  of  12.     23.     39.     24. 

3.  Find  the  second  power  of  47.     51.     29.     34. 

4.  What  is  the  square  of  15  ?     33  ?     24  ?     36  ?    25  ? 

5.  What  is  the  cube  of  28?     45?     18?     21?     41? 

6.  What  is  the  third  power  of  f  ?     A  ns.  -f-  X  -f  X  -f-  =  iff- 

7.  What  is  the  cube  of  |?     Off?    f?    T6T? 

8.  What  is  the  fourth  power  off?     Cube  of 

Find  the  value  of  the  following : 


9.  15*. 
10.  253. 
11.  303. 

12.  .05*. 
13.  .005s. 
14.  2.052. 

15.  (if)2. 

16.  (M)2- 
17.  (4i)8. 

18.  (25|)V 
19.  (3.001)2. 
20.  (4.500})2. 

21.  Eaise  10  to  the  fourth  power;   8  to  the  third  power; 
3  to  the  6th  power. 


INVOLUTION. 


317 


505.  To  find  the  square  of  a  number  in  terms  of  its 
parts. 

1.  Find  the  square  of  35  in  terms  of  its  tens  and  units. 

PROCESS.  ANALYSIS. — If  we  square  35  or 

multiply    35    by   itself    and    write 


35 
35 

25  =  w2 
15 
15 
9         =f 


every  step  in  the  process,  we  shall 
have  for  the  first  product  25,  or  the 
square  of  the  units,  for  the  next  two 
products  15  tens,  or  two  times  the 
product  of  the  tens  and  units,  and 
for  the  third  product  9  hundreds  or 
the  square  of  the  tens.  Hence, 


506.  PRINCIPLE.  —  The  square  of  any  number  consisting  of 
tens  and  units,  is  equal  to  the  tens2  -f-  2  times  the  tens  X  the  units 
-\-  the  units2. 

Thus,  25  =  20  +  5,  and  25  2  =  202  +  2  (20  X  5)  .+  5  2. 

The  above  principle  is  true  into  whatever  two  parts  the  number 
may  be  separated,  and  the  principle  stated  in  general  terms  would 
be,  the  square  of  any  number  consisting  of  two  parts  is  equal  to  the 
first  part  2  +  2  times  the  first  part  X  the  second  +  second  part  2. 

Thus,  14  =  8  +  6,  and  142  =  82  +  2  (6X8) 


Express  in  terms  of  their  tens  and  units  the  square  of  the 
following  numbers: 


2. 

54. 

5. 

47. 

8.  74. 

11. 

39. 

3. 

71. 

6. 

89. 

9.  95. 

12. 

44. 

4. 

68. 

7. 

26. 

10.  82. 

13. 

67, 

14. 

Square 

16 

by 

squaring 

its 

parts    9 

and  7. 

15. 

Square 

20 

by 

squaring 

its 

parts  12 

and  8. 

16. 

Square 

32 

by 

squaring 

its 

parts  30 

and  2. 

17. 

Square 

13 

by 

squaring 

its 

parts    7 

and  6. 

18. 

Square 

26 

by 

squaring 

its 

parts    9 

and  17. 

19. 

Square 

17 

by 

squaring 

its 

parts    8 

and  9. 

318  INVOLUTION. 

507.   To  find  the  cube  of  a  number  in  terms  of  its 
parts. 

1.  Find  the  cube  of  35  in  terms  of  its  parts. 


25 

*•=  s 


ANALYSIS. — By  multiplying  the  second  power  expressed  as  in 
Art.  5O5,  by  35,  and  writing  every  step,  we  shall  have  the  cube  of 
the  tens,  plus  the  product  of  three  times  the  square  of  the  tens  multi- 
plied by  the  units,  plus  the  product  of  three  times  the  tens  multiplied 
by  the  square  of  the  units,  plus  the  cube  of  the  units.  Hence, 

508.  PRINCIPLE. — The  cube  of  any  number  consisting  of  tens 
and  units  is  equal  to  the  tens  s  +  3  times  the  tens  2  X  the  unite 
+  3  times  the  tens  X  the  units2  +  the  units*. 

Thus,  25  =  20  +  5,  and  253  =  203  +  3  (202  X  5)  +  3  (20  X  52)  +  53. 

The  above  principle  may  be  stated  in  general  terms  thus:  The  cube 
of  any  number  when  separated  into  two  parts  is  equal  to  the  first  part 3 
+  3  times  the  first  part  2  X  second  part  +  3  times  the  first  part  multi- 
plied by. the  second  part  <J  +  the  second  part  3. 

Express  in  terms  of  their  tens  and  units  the  cube  of  the 
following  numbers: 


2.  26. 

3.  31. 


5.  42. 

6.  27. 


4.  28.  7.  36. 


8.  38.  11.  52. 

9.  39.  12.  64. 
10.  54                 13.  66. 


TVOLUTIOTT 


509.  1.  What  are  the  factors  of  36?  What  are  the  two 
equal  factors  of  36?  Of  49?  Of  81? 

2.  What  number  used  three  times  as  a  factor  will  produce 
27?  64?  125?  216? 


DEFINITIONS. 

510.  A  Root  of  a  number  is  one  of  the  equal  factors  of 
the  number. 

Thus,  4  is  a  root  of  16,  because  it  is  one  of  two  equal  factors. 

Roots  are  named  in  a  manner  similar  to  powers.  Thus, 
one  of  two  equal  factors  of  a  number  is  the  second,  or  square 
root;  one  of  three  equal  factors,  the  third,  or  cube  root;  one 
of  four  equal  factors,  the  fourth  root,  etc. 

511.  Evolution  is  the  process  of  finding  roots  of  num- 
bers. 

512.  The  Radical,  or  Root  Sign,  is  j/.     When 
placed  before  a  number  it  shows  that  its  root  is  to  be  found. 

When  no  figure  or  index  is  written  in  the  opening  of  the 
radical  sign,  the  square  root  is  indicated;  if  the  figure  3  is 
placed  there,  as  ^,  the  cube  root  is  indicated;  if  4,  as  \/ , 
the  fourth  root;  etc. 

513.  A  Perfect  Power  is  a  number  whose  root  can  be 
found. 

(319) 


320  EVOLUTION. 

514.  An  Tmperfect  Power  is  a  number  whose  root 
can  not  be  found  exactly. 


EVOLUTION  BY  FACTORING. 

515.  1.  What  is  the  square  root  of  1225  ? 

PROCESS.  ANALYSIS. — Since  the 

5)1  225  square  root  of  a  number 

r^  n  A  r  is   one  of    its  two  equal 

factors,  we  may  find  the 

7)49  square   root  of    1225    by 

7)7     1/1225  =  5  X  7  =  35        separating     it     into     its 

prime  factors,  and  find- 
ing the  product  of  the  numbers  forming  one  of  the  two  equal  sets  of 
factors.  The  prime  factors  are  5,  5,  7  and  7,  and  5  and  7  form  one  of 
the  two  equal  sets.  Therefore  their  product,  35,  is  the  square  root 
of  1225. 

RULE. — Separate  the  numbers  into  their  prime  factors.  Ar- 
range these  factors  into  two,  three,  four,  or  any  number  of  sets 
containing  the  same  factors,  according  as  the  second,  third,  fourth, 
or  any  root  is  to  be  found. 

The  product  of  the  factors  which  form  a  set  will  be  the  root. 

This  method  is  valuable  only  when  the  numbers  whose  roots  are 
sought  are  perfect  powers. 

2.  Find  the  square  root  of    144.     256.       324.     576. 

3.  Find  the  cube    root  of      64.     512.     4096.     13824. 

4.  Find  the  fourth  root  of  1296.     The  fifth  root  of  248832. 

ORDERS  OF  UNITS  IN  POWERS  AND  ROOTS. 

516.  1.  How  does  the  number  of  figures  which  express  the 
square  of  units,  compare  with  the  number  expressing  units? 

2.  How  does  the  number  of  figures  required  to  express  the 
cube  of  units  compare  with  the  number  expressing  units? 


EVOLUTION.  321 

3.  Write  the  numbers,  10,  99,  100,  999,  1000,  and  under 
them  their  second  and  third  powers. 

4.  How  does  the  number  of  figures  required  to  express  the 
second  powers  compare  with  the  number  of  figures  required 
to  express  the  given  numbers? 

5.  How  does  the  number  of  figures  required  to  express  the 
third  powers  compare  with  the  number  of  figures  required  to 
express  the  given  numbers? 

517.  PRINCIPLES. — 1.  The  square  of  a  number  is  expressed 
by  twice  as  many  figures  as  the  number  itself  or  one  less. 

2.  The  cube  of  a  number  is  expressed  by  three  times  as  many 
figures  as  the  number  itself  or  one  or  two  less. 


EXEItC  ISE  S. 

518.  Tell  by  referring  to  the  principles,  how  many  figures 
there  are  in  the  following : 

1.  In  the\square  of  21.  Of  15.  Of  115.  Of  4156. 

2.  In  the  cube     of  19.  Of  25.  Of  316.  Of  6184. 

3.  In  the  square  of  35.  Of  29.  Of  584.  Of  8196. 

4.  In  the  cube     of  59.  Of  67.  Of  999.  Of  9999. 

5.  How  many  figures  or  orders  of  units  are  there  in  a  num- 
ber if  the  second  power  of  it  is  expressed  by  4  figures?     By  7 
figures  ?     By  9  figures  ? 

6.  How  many  figures  or  orders  of  units  are  there  in  a  num- 
ber if  the  third  power  of  it  is  expressed  by  6  figures  ?     By  8 
figures?     By  12  figures?     By  11  figures?     By  21  figures? 
By  25  figures? 

519.  PRINCIPLES. — 1.   The  orders  of  units  in  the  square  root 
of  a  number  correspond  to  the  number  of  periods  of  two  figures  each 
into  which  Hie  number  can  be  separated,  beginning  at  units. 

21 


322 


EVOLUTION. 


2.  The  orders  of  units  in  the  cube  root  of  a  number  correspond 
to  the  number  of  periods  of  three  figures  each  into  which  the  number 
can  be  separated,  beginning  at  units. 


SQUARE  ROOT. 

520.    1.  What  is  the  square  root  of  576,  or  what  is  the 
side  of  a  square  whose  area  is  576  square  units  ? 


IST  PROCESS. 

576(20 
2  0 2  =  400    _4 

2X20  =  40)176    24 
(40  +  4)  X4  =  176 


B--- 


ANALYSIS.  —  According  to  Prin. 
1,  Art.  519,  the  orders  of  units 
in  the  square  root  of  any  number 
may  be  determined  by  separating 
the  number  into  periods  of  two 
figures  each,  beginning  at  units. 
Separating  576  thus,  there  are 
found  to  be  two  orders  of  units  in 
the  root,  or  it  is  composed  of  tens 
and  units.  Since  the  square  of 
tens  is  hundreds,  5  hundreds  must 
be  the  square  of  at  least  2  tens.  2 
tens  or  20  squared  is  400,  and  400 
subtracted  from  576  leaves  176, 
therefore  the  root  20  must  be  in- 
creased by  such  an  amount  as  will 
exhaust  the  remainder. 

The  square  (A)  already  formed 
from  the  576  square  units  is  one 
whose  side  is  20  units,  but  inas- 
much as  the  number  of  units  was 
not  exhausted,  such  additions  must 
be  made  to  the  square  that  they 
will  exhaust  the  units  and  keep  the 
figure  a  square.  The  necessary  ad- 
ditions are  two  equal  rectangles 
B  and  C,  and  a  small  square  D. 
Since  the  square  D  is  small,  the  area  of  the  rectangles  B  and  C 


B 


D 


20 


g 


SQUAEE   ROOT.  323 

is  nearly  176  units.  The  area,  176  units,  divided  by  the  length  of 
the  rectangles,  will  give  the  width,  which  is  4  units.  The  width  of 
the  additions  is  4  units,  and  the  entire  length,  including  the  small 
square,  is  44  units ;  therefore  the  area  of  all  the  additions  is  4  times 
44  units,  or  176  square  units,  which  is  equal  to  the  entire  number 
of  units  to  be  added.  Therefore  the  side  of  the  square  is  24  units, 
or  the  square  root  of  the  number  is  24. 

2o  PROCESS.  ANALYSIS.— In  the  same  manner 

5*7  6  (24     as  Before,  the  root  of  this  number  is 

,2 02 4  shown  to  consist  of  tens  and  units. 

The  tens  can  not  be  greater  than 


2£  — 40 
u=    4 


2t  +  u  —  4  4 


1  '  b  2;  therefore  we  write  2  tens  for  a 

partial   root.     Squaring    and   sub- 
176  tracting,   there   is    a   remainder  of 


176,  which  is  composed  of  2  times 
the  tens  X  units  +  units2,  Art.  506.  Since  2  times  the  tens  multi- 
plied by  the  units  is  much  greater  than  the  units  squared,  176  is  nearly 
two  times  the  tens  multiplied  by  the  units.  Therefore  if  176  is  di- 
vided by  twice  the  tens,  or  40,  the  quotient  will  be  approximately 
the  units  of  the  root.  Dividing,  the  units  are  found  to  be  4. 

Since  the  tens  are  to  be  multiplied  by  the  units,  and  the  units  are 
to  be  multiplied  by  the  units  or  squared,  and  these  results  are  to  be 
added,  to  abridge  the  process  the  units  are  added  to  twice  the  tens  and 
the  sum  multiplied  by  the  units.  Thus,  40  +  4  is  multiplied  by  4, 
making  176.  Therefore  the  square  root  of  576  is  24. 

When  the  root  consists  of  more  than  two  orders  of  units, 
the  process  for  solving  is  similar  to  that  already  given. 

2.  Find  the  square  root  of  137641. 

1ST  PROCESS.  2D  PROCESS. 

13-76-41  (300  13-76-41(371 

9  00  00     70  9 


300x2—600+70—670)4  76  41 1          67)476 

46900  371  469 


370x2—740+1=741  )7  41  741)741 

7  41  741 


324  EVOLUTION. 

In  the  first  process  the  steps  are  given  with  considerable 
fullness,  while  in  the  second  process  all  steps  are  abbreviated 
as  much  as  possible. 

RULE. — Separate  the  number  into  periods  of  two  figures  each, 
beginning  at  units. 

Find  the  greatest  square  in  the  left-liand  period,  and  write  its 
root  for  the  first  figure  of  the  required  root. 

Square  this  root  and  subtract  the  result  from  "the  left -hand 
period,  and  annex  to  the  remainder  the  next  period  for  a  dividend. 

Double  the  root  already  found  for  a  trial  divisor,  and  by  it 
divide  the  dividend,  disregarding  the  right-hand  figure.  The 
quotient  or  quotient  diminished  will  be  the  second  figure  of  the  root. 

Annex  to  the  trial  divisor  for  a  complete  divisor,  the  figure  last 
found,  multiply  this  divisor  by  the  last  figure  of  the  root  found, 
subtract  the  product  from  the  dividend,  and  to  the  remainder  annex 
the  next  period  for  the  next  dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used 
thus.  The  result  will  be  ike  square  root  sought. 

1.  When  the  number  is  not  a  perfect  square,  annex  periods  of 
ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  into  periods  of  two  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 

3.  The  square  root  of   a  common  fraction  is  found   by  extracting 
the  square  root  of  both  numerator  and  denominator  separately,  or 
by  reducing  it  to  a  decimal  and  then  extracting  its  root. 

Extract  the  square  root  of  the  following : 


3.  2809. 
4.  3969. 
5.  4356. 
6.  9216. 

7.  70756. 
8.  118336. 
9.  674041. 
10.  784996. 

11.  938961. 
12.  5875776. 
13.  12574116. 
14.  30858025. 

15.  Find  the  value  of  1/222784;  T/11390625. 


16.  Find  the  value  of  1/.763876;  1/.30858025. 


SQUARE   ROOT.    ,  325 

17.  What  is  the  square  root  of  .093636? 

18.  What  is  the  square  root  of  .099225? 

19.  What  is  the  square  root  of  f 

20.  Extract  the  square  root  of  £ 

21.  Extract  the  square  root  of  | 

22.  Extract  the  square  root  of  ^. 

23.  Extract  the  square  root  of  f . 

24.  Extract  the  square  root  of  f. 

25.  Extract  the  square  root  of  .9. 

APPLICATIONS  OF  SQUARE  ROOT. 

521*  To  find  the  side  of  a  square  when  its  area  is 
given. 

Since  the  area  of  a  square  is  the  product  of  two  equal 
factors  which  represent  its  sides,  the  sides  may  be  found 
by  extracting  the  square  root  of  the  number  expressing  its 
area. 

1.  What  is  the  side  of  a  square  whose  area  is  625  square 
feet? 

2.  What  is  the  side  of  a  square  whose  area  is  2025  square 
rods? 

3.  A  rectangle  whose  area  is  5408  square  feet  is  composed 
of  two  equal  squares.     What  is  the  length  of  its  sides  ? 

4.  A  man  owns  50  acres  of  land  in  two  square  fields,  one 
of  which  contains  4  times  as  much  area  as  the  other.     How 
many  rods  of  fence  will  be  needed  to  fence  both  fields  if 
they  are  not  adjacent? 

5.  The  length  of  a  rectangular  field  containing  20  acres  is 
twice  its  width.     What  is  the  distance  around  it? 

6.  If  it  cost  $572   to  inclose  with  a  fence  a  field  that  is 
72  rods  long  and  32  rods  wide,  how  much  less  will  be  the 
cost  of  inclosing  a  square  field  containing  the  same  area? 


326 


EVOLUTIOK. 


522.   To  find  any  side  of  a  right-angled  triangle  when 
the  other  sides  are  given. 


Triangle 


Eight  angle 


Base 


523.  A  Triangle  is  a  figure  which  has  three  angles 
and  three  sides. 

524.  A  Right  Angle  is  the  angle  formed  when  one 
line  is  drawn  perpendicular  to  another. 

525.  A  Hight -angled  Triangle  is  a  triangle  which 
has  a  right  angle. 

526.  The  Hypotenuse  of  a  right-angled  triangle  is  the 
side  opposite  the  right  angle. 

527.  The  Base  of  a  triangle  is  the  side  on  which  it  is 
assumed  to  stand. 

528.  The  Perpendicular  is  the  side  which  forms  a 

right-angle  with  the  base. 

The  relation  of  the  squares  de- 
scribed upon  the  sides  of  a  right- 
angled  triangle  is  expressed  thus: 

529.   PRINCIPLES.  — 1.  The 

square  described  upon  the  hypote- 
nuse of  a  right-angled  triangle  is 
equal  to  the  sum  of  the  squares  on 
the  other  two  sides. 

2.  The  square  on  either  of  the 
other  sides  of  a  right-angled  triangle  is  equal  to  the  square  on  the 
hypotenuse  diminished  by  the  square  on  the  other  side. 


SQUARE   ROOT.  327 

When  the  number  of  square  units  in  the  surface  of  any  square 
figure  is  known,  its  side  may  be  found  by  extracting  the  square 
root  of  the  number  according  to  the  preceding  case. 

530,  1.  The  base  of  a  right-angled  triangle  is  8  feet  and  the 
perpendicular  6  feet,  what  is  the  hypotenuse  ? 

PROCESS.  ANALYSIS. — Before  we  can  determine  the 

£2     ,    ~2 100  length  of  the  hypotenuse  when  the  sides  are 

given  we  must  find  the  area  of  a  square  de- 

l/  1  0  0  =  1  0 .  scribed  upon  it.     The  square  described  upon 

it  is  equal  to  the  sum  of  the  squares  upon 

the  other  two  sides,  or  the  sum  of  82+  62,  which  is  100.  Since  the  area 
of  the  square  described  upon  the  hypotenuse  contains  100  square  units, 
the  length  of  the  side  is  the  square  root  of  100  or  10. 

To  find  the  hypotenuse. 

EULE. — Extract  the  square  root  of  the  sum  of  the  squares  of  the 
other  two  sides. 

To  find  the  base  or  perpendicular. 

EULE. — Extract  the  square  root  of  the  difference  of  the  squares 
on  the  hypotenuse  and  the  other  side. 

2.  The  base  of  a  right-angled  triangle  is  15  feet  and  the  per- 
pendicular is  20  feet.     What  is  the  hypotenuse  ? 

3.  The  base  of  a  right-angled  triangle  is  40  feet  and  the 
hypotenuse  is  120  feet.     What  is  the  perpendicular? 

4.  The  perpendicular  of  a  right-angled  triangle  is  30  feet 
and  the  hypotenuse  is  50  feet.     What  is  the  base? 

5.  A  tree  150  feet  high,  standing  upon  the  bank  of  a  stream, 
was  broken  off  125  feet  from  the  top,  and  falling  across  the 
stream   the  top  just  reached  the  other  shore.     What  was  the 
width  of  the  stream? 

6.  Two  steamers  start  from  the  same  point,  one  going  due 
north  at  the  rate  of  15  miles  an  hour,  and  the  other  going  due 


328  EVOLUTION. 

west  at  the  rate  of  18  miles  an  hour.    How  far  apart  were  they 
at  the  end  of  6  hours  ? 

7.  A  rectangular  park,  whose  sides  are  respectively  45  rods 
and  60  rods  in  length  has  a  walk  crossing  it  from  corner  to 
corner.     How  long  is  the  walk? 

8.  A  certain  assembly  room  is  100  feet  in  length,  60  feet  in 
width,  and  26  feet  in  height.     What  is  the  distance  from  a 
lower  corner  to  the  upper  opposite  corner  ? 

9.  Two  buildings  standing  opposite  each  other  are  respect- 
ively 60  feet  and  80  feet  high.     A  ladder  125  feet  long  placed 
at  a  certain  distance  from  the  base  of  each  just  reaches  the 
top  of  each.     How  far  apart  are  the  buildings? 

10.  The  distance  from  the  base  of  a  building  to  a  pole  is 
145  feet,  and  a  string  225  feet  long  attached  to  the  top  of  the 
pole  just  reaches  the  base  of  the  building.    What  is  the  height 
of  the  pole  ? 

11.  A  person  who  wished  to  ascertain  the  exact  height  of 
St.  Paul's  Cathedral  in  London,  England,  learned  by  inquiry 
that  a  rope  extending  from  the  top  of  the  cross  to  a  point 
300  feet  from  the  center  of  the  circular  pavement  under  the 
dome  was  488  feet  10^  inches  long.     If  these  data  were  cor- 
rect, what  is  the  height  of  St.  Paul's  ? 


SIMILAR  FIGURES. 

531.  Similar  Figures  are  such  as  are  of  the  same 
form,  and  differ  from  each  other  only  in  size. 

The  truth  of  the   following  principles   can  be  shown  by 
geometry : 

532.  PRINCIPLES. — 1.  Similar  surfaces  are  to  each  other  as 
the  squares  of  their  corresponding  dimensions.     Hence, 

2.    The  corresponding  dimensions  of  similar  surfaces  are  to  each 
other  as  the  square  roots  of  their  areas. 


SQUABE    ROOT.  329 

1.  If  the  area  of  a  triangle  whose  base  is  16  rods,  is  128 
square  rods,  how  many  square  rods  are  there  in  the  area  of  a 
similar  triangle  whose  base  is  12  rods  ? 

PROCESS.  ANALYSIS. — Since  the  areas  of  similar 

1 9«  1  ft2      1 92  figures  are  to  each  other  as  the  squares 

OT>        of  their  like  dimensions,  the  area  of  the 

128  :  x  : :  256  :  144  first  triangle  (128  sq.  rVl.)  will  be  to  the 

x  __  7 2  Sq    r(J  area  of  the  second  triangle  ( x )  as  the 

square  of  the  side  of  the  first  triangle 

(162)  is  to  the  square  of  the  side  of  the  second  triangle  (122).    Solving 
the  proportion,  the  area  is  72  sq.  rd. 

2.  If  the  area  of  a  circle,  whose  diameter  is  2  feet,  is  6.2832 
sq.  ft.  what  will  be  the  area  of  a  circle  whose  diameter  is  4  feet? 

3.  If  the  side  of  a  rectangular  field  containing  25  acres  is 
40  rods,  what  is  the  side  of  a  similar  field  containing  10  acres? 

ANALYSIS. — Since  the  corresponding  dimensions  of  similar  surfaces 
are  to  each  other  as  the  square  roots  of  their  areas: 

!/25  :  v/IO"  ::  40  :  x,  or  5  :  y'IO  ::  40  :  x. 

Extracting  the  square  root  of  10  and  solving  the  proportion,  xt  or 
the  corresponding  side,  is  25.296  rd. 

4.  If  the  side  of  a  square  field  containing   40   acres  is  80 
rods,  what  will  be  the  side  of  a  similar  field  whose  area  is  25 
acres? 

5.  If  the  area  of  a  circle  whose  diameter  is  20  feet  is  314.16 
square  feet,  what  is  the  diameter  of  a  circle  whose  area  is 
113. 0976  square  feet? 

6.  A  farmer  has  two  rectangular  fields  similar  in  form: 
one,  whose  length  is  120  rods  and  whose  breadth  is  12  rods, 
contains  9  acres,  the  other  contains  6J  acres.     What  are  its 
length  and  breadth? 

7.  A  horse  tied  to  a  stake  by  a  rope  7.13  rods  long  can 
graze  upon  just  1  acre  of  ground.     How  long  must  the  rope 
be  that  he  may  graze  upon  5  acres? 


330 


EVOLUTION. 


1ST  PROCESS. 


208  =. 

3X20*  =  1200 


13-824(20+4=24 
8000 


3  X  4  X  20  = 
4?  = 


240 
16 


1456 


5824 


5824 


CUBE   ROOT. 

533.    1.  What  is  the  cube  root  of  13824,  or  what  is  the 
edge  of  a  cube  whose  solid  contents  are  13824  units? 

ANALYSIS. — Accord- 
ing to  Pr.  2,  Art.  51.9, 
the  orders  of  units  in 
the  cube  root  of  any 
number  may  be  de- 
termined from  the 
number  of  periods  ob- 
tained by  separating 
the  number  into  pe- 
riods, containing  three 

figures  each,  beginning  at  units.     Separating  the  given  number  thus, 
there  are  two  periods,  or  the  root  is  composed  of  tens  and  units. 

The  tens  in  the  cube  root  of 
the  number  can  not  be  greater 
than  2,  for  the  cube  of  3  tens  is 
27000.  2  tens,  or  20  cubed,  are 
8000,  which,  subtracted  from 
13824,  leave  5824;  therefore 
the  root,  20,  must  be  increased 
by  a  number  such  that  the 
additions  will  exhaust  the  re- 
mainder. 

The  cube  (A)  already  formed 
from  the  13824  cubic  units  is 
one  whose  edge  is  20  units. 
The  additions  to  be  made, 
keeping  the  figure  formed  a 
perfect  cube,  are  3  equal  rect- 
angular solids,  B,  C  and  D; 
3  other  equal  rectangular 
solids,  E,  F  and  G;  and  a 
small  cube,  H.  Inasmuch  as 
the  solids,  B,  C  and  D,  com- 


CUBE   ROOT. 


331 


prise  much  the  greatest  part  of  the  additions,  their  solid  contents 
will  be  nearly  5824  cubic  units,  the  contents  of  the  addition. 

Since  the  cubical  contents 
of  these  three  equal  solids  are 
nearly  equal  to  5824  units,  and 
the  superficial  contents  of  a 
side  of  each  of  these  solids  are 
20  X  20,  or  400  square  units, 
if  we  divide  5824  by  3  times 
400,  or  1200,  since  there  are  3 
equal  solids,  we  shall  obtain 
the  thickness  of  the  addition, 
which  is  4  units. 

Since  all  the  additions  have 
the  same  thickness,  if  their  su- 
perficial contents,  or  area  of 
each  side,  are  multiplied  by 
4,  the  result  will  be  the  solid 
contents  of  these  additions. 

Besides  the  larger  additions 
there  are  three  others,  E,  F, 
and  G,  that  are  each  20  units 
long  and  4  units  wide,  or  whose 
sides  have  an  area  of  80  units  each,  and  the  area  of  all  240  units,  and 
a  small  cube  whose  sides  have  an  area  of  16  units.  The  sum  of  these 
areas,  1456,  multiplied  by  4,  the  thickness  of  the  additions,  gives  the 
solid  contents  of  the  additions,  which  are  5824  units. 

Therefore  the  edge  of  the  cube  is  24  units  in  length,  or  the  cube 
root  of  13824  is  24. 


2D  PROCESS. 

3J2  =  202  X  3  =  1200 
(20  X  4)  X  3  =   240 
tia  =  4x4=      16 

13,824(24 
8000 

ANALYSIS.  —  In 
the  same  manner 
as  before,  it  may 
be     shown     that 
the   root   of    the 
number  contains 
only     tens     and 
units.     The  lens 
can  not  be  great- 
er than  2,  for  3 

5824 

5t2  +  Stu  +  n2  =  U5Q 

5824 

332 


EVOLUTION. 


tens  cubed  would  equal  27000.  Cubing  and  subtracting,  there  is  left 
5824,  which  is  composed  of  three  times  the  tens2  X  the  units  +  3  times 
the  tens  X  the  units2  +  the  units3,  Art.  507. 

Since  3  times  the  tens2  is  much  greater  than  3  times  the  tens  X  the 
units2  +  the  units3,  5824  is  a  little  more  than  3  times  the  tens2  X  the 
units.  If,  then,  5824  is  divided  by  3  times  the  tens2,  or  1200,  the  trial 
divisor,  the  quotient  4,  will  be  approximately  the  units  of  the  root. 

Since  5824  is  equal  to  the  sum  of  3  times  the  tens2  multiplied  by 
the  units,  3  times  the  tens  multiplied  by  the  units2  and  the  units3,  the 
process  may  be  shortened  by  adding  together  3  times  the  tens2,  3  times 
the  tens  X  the  units  and  the  units2,  and  multiplying  this  sum,  1456, 
by  the  units,  4.  The  product  is  5824,  which,  subtracted  from  the  num- 
ber, leaves  no  remainder. 

When  the  root  consists  of  more  than  two  orders  of  units 
the  processes  and  analyses  are  similar  to  those  already  given. 

2.  What  is  the  cube  root  of  48228544? 


3X 
3X300 

300s 
3002 
X60 
602 

=  270000 
=   54000 
=     3600 

48 
27 

•228 
000 

•544 
000 

300 
60 
4 

21 

,  s 

19 

228 
656 

544 
000 

364 

327600 

3  X  3602  =  388800 
3x360x4=     4320 

42= 16 

393136 


1  572  544 


1  572  544 


RULE. — Separate  the  number  into  periods  of  three  figures  each, 
beginning  at  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write  its 
root  for  the  first  part  of  the  required  root. 

Cube  this  root,  subtract  the  result  from  the  left-hand  period, 
and  annex  to  the  remainder  the  next  period  for  a  dividend. 

Take  3  times  the  square  of  the  root  already  found  for  a  trial 


CUBE   ROOT. 


333 


divisor,  and  by  it  divide  the  dividend.  The  quotient  or  the  quo- 
tient diminished  will  be  the  second  part  of  the  root. 

To  this  trial  divisor  add  3  times  the  product  of  the  first  part 
of  the  root  by  the  second  part,  and  also  the  square  of  the  second 
part.  Their  sum  will  be  the  entire  divisor. 

Multiply  the  entire  divisor  by  the  second  part  of  the  root  and 
subtract  the  product  from  the  dividend. 

Continue  thus  until  all  the  figures  of  the  root  have  been  found. 

1.  When  there  is  a  remainder,  after  subtracting  the  last  product 
annex  decimal  ciphers,  and  continue  the  process.    The  figures  of  the 
root  obtained  will  be  decimals. 

2.  Decimals  are  pointed  off  into  periods  of  three  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 

3.  The  cube  root  of  a  common  fraction  is  found  by  extracting  the 
cube  root  of    both   numerator    and    denominator    separately,  or   by 
reducing  it  to  a  decimal  and  then  extracting  its  root. 

Extract  the  cube  root  of  the  following : 


3.  74088. 
4.  262144. 
5.  166375. 

6.  704969. 
7.  185193. 
8.  250047. 

9.  5545233. 
10.  2000376. 
11.  153990656. 

12.  What  is  the  cube  root  of  2  to  3  decimal  places? 

13.  What  is  the  cube  root  of  9  to  4  decimal  places? 

14.  What  is  the  cube  root  of  .27  ?     Of  .64? 

15.  What  is  the  cube  root  of  f  ?     f  ? 


APPLICATIONS  OF  CUBE  ROOT. 

534.  1.  What  is  the  length  of  the  edge  of  a  cubical  box 
that  contains  91125  cubic  feet? 

2.  What  are  the  dimensions  of  a  cubical  box  that  con- 
tains as  much  as  a  rectangular  box,  that  is  2  feet  8  inches 
long,  2  feet  3  inches  wide,  and  1  foot  4  inches  deep  ? 


S34  EVOLUTION. 

3.  What  is  the  depth  of  a  cubical  cistern  whose  contents 
are  2197  cubic  feet? 

4.  What  must  be  the  depth  of  a  cubical  bin  that  will 
contain  1000  bushels? 

5.  What  must  be  the  depth  of  a  cubical  cistern  that  will 
hold  300  barrels  of  water? 

6.  A  bin    that  contains    2000    bushels    of  grain,   is  just 
twice  as  long  as  it  is  wide  or  high.     What  is  its  length  ? 

7.  What  is  the  depth   of  a  cubical  box  that  will  hold  a 
bushel  ? 

8.  What  is  the  depth  of  a  cubical  box  that  will  hold  a 
barrel  of  water  (31-J-  gal.)  ? 

9.  How  much  will  it  cost,  at  30  cents  per  sq.  yd.,  to  plaster 
the  bottom  and  sides  of  a  cubical  cistern  that  will  hold  300 
barrels  ? 

10.  A  miller  wishes  to  make  a  wagon-box  large  enough  to 
hold  100  bushels,  having  the  length  3  times  the  width  and 
height.     What  will  be  its  dimensions? 

11.  Which  has   the  greater  surface,  a  cube  whose   solid 
contents  are  13824  cubic  feet,  or  a  rectangular  solid  having 
the  same  solidity,  whose  height  is  half  its  length  and  whose 
width  is  three-fourths  its  length?     How  much? 

12.  If  a  cubic  metre  contains  61026.048  cubic  inches,  what 
is  the  length  of  a  linear  metre  ? 

SIMILAR  FIGURES. 

535.  The  truth  of  the  following  principles  can  be  shown 
by  geometry: 

PRINCIPLES. — 1.  Similar  solids  are  to  each  other  as  the  cubes 
of  their  like  dimensions.  Hence, 

2.  The  corresponding  dimensions  of  similar  solids  are  to  each 
other  as  the  cube  roots  of  their  volumes. 


CUBE   HOOT.  335 

1.  If  a  globe  4  inches  in  diameter  weighs  8  lb.,  what  will 
be  the  diameter  of  a  similar  one  that  weighs  125  lb.  ? 

PROCESS.  ANALYSIS. — Since   the  corre- 

A.          •       a/S"  •    3/T9^    n\         spending  dimensions  of  similar 

solids   are   proportional    to    the 

4  :  x  :  :       2  :  5    (2)         cube  roots  of  these  volumes,  we 

x  or  diameter  is  10  in.  have  the.  diameter  of  the  smaller 

globe  4  inches  :  the  diameter  of 

the  larger  globe  x  :  :  the  cube  root  of  the  weight  of  the  smaller  globe 
•ty  8  :  the  cube  root  of  the  weight  of  the  other  globe  ^'125.  (1).  Ex- 
tracting the  cube  root  of  8  and  125,  and  we  have  Prop.  (2).  Whence 
solving,  the  diameter  is  10  inches. 

2.  If  a  ball  5  ft.  in  diameter  weighs  800  lb.,  what  will 
be  the  diameter  of  a  similar  ball   which  weighs  3  T.  4  cwt.  ? 

3.  If  a  globe  of  gold  1  inch  in  diameter  is  worth  $125, 
what  will  be  the  value  of  one  3  inches  in  diameter? 

4.  If  a  cubical  bin  8  ft.  long  will  hold  411.42  bu.,  what 
must  be  the  dimensions  of  a  similar  bin,  that  will  hold  1000 
bushels  ? 

5.  A  ball  3  feet  in  diameter  weighed  2000  lb.     What  will 
be  the  diameter  of  one  that  weighs  1000  lb.  ? 

6.  The  dimensions  of  a  cubical  bin  were  such  that  it  would 
contain  1000  bushels  of  wheat.     How  would  the  dimensions 
of  a  similar  bin  that  would  hold  8000  bushels  compare  with 
the  dimensions  of  such  a  bin? 

7.  The  diameters  of  two  spheres  are  respectively  4  and 
12   inches.      How   many  times    the    smaller   sphere    is   the 
larger? 

8.  Three  women  own  a  ball  of  yarn  4  inches  in  diameter. 
How  much  of  the  diameter  of  the  ball  must  each  wind  off. 
so  that  they  may  share  equally? 

9.  A  stack  of  hay  in  the  form  of  a  pyramid  12  ft.  high, 
contained  8  tons.     How  high  must  a  similar  stack  be,  that 
it  may  contain  60  tons? 


536.    1.  How  does  each  of  the  numbers  2,  4,  6,  8,  10,  12, 
compare  with  the  number  that  follows  it? 

2.  How  may  each  of  the  numbers  4,  6,  8,  etc.,  be  obtained 
from  the  one  that  precedes  it? 

3.  How  does  each  of  the  numbers  2,  5,  8,  11,  14,  17,  com- 
pare with  the  number  that  follows  it?     How  with  the  one 
that  precedes  it? 

4.  Write  in  succession   some  numbers  beginning  with  3 
having  a  common  difference  of  2. 

5.  Write  a  series  of  numbers  beginning  at  4,  and  having  a 
common  difference  of  4. 

6.  Write  a  series  of  numbers  beginning  with  25,  and  de- 
creasing by  the  common  difference  4. 

7.  How  does  each  of  the  numbers  2,  4,  8,  16,  32,  etc.,  com- 
pare with  the  one  that  follows  it  ?     How  may  each  be  obtained 
from  the  one  that  precedes  it? 

8.  Write  a  series  of  numbers  beginning  with  2  and  increas- 
ing by  a  common  multiplier  3. 

9.  Write  a  series  of  numbers  beginning  with  5,  and  increas- 
ing by  a  common  multiplier  5. 


DEFINITIONS. 

537.  A  Series  of  numbers  is  numbers  in  succession,  each 
derived  from  the  preceding  according  to  some  fixed  laws. 

(336) 


ARITHMETICAL   PROGRESSION.  337 

538.  The  first  and  last  terms  of  a  series  are  called  the 
extremes,  the  intervening  terms  the  means. 

Thus,  in  the  series  2,  4,  6,  8,  10,  the  numbers  2  and  10  are  the  ex- 
tremes and  the  others  are  the  means. 

539.  An  Ascending  Series  is  one  in  which  the  num- 
bers increase  regularly  from  the  first  term. 

Thus,  2,  5,  8,  11,  14,  17,  20,  etc.,  is  an  ascending  series. 

540.  A  Descending  Series  is  one  in  which  the  num- 
bers decrease  regularly  from  the  first  term. 

Thus,  48,  24,  12,  6,  3,  is  a  descending  series. 


ARITHMETICAL  PROGRESSION. 

541.  An  Arithmetical  Progression  is  a  series  of 
numbers  which  increase  or  decrease  by  a  constant  common 
difference. 

Thus,  5,  9,  13,  17,  21,  etc.,  is  an  arithmetical  progression  of  which 
the  common  difference  is  4. 

542.  1.  The  first  term  of  an  arithmetical  series  is  3  and 
the  common  difference  is  2.     What  is  the  7th  term? 

PROCESS.  ANALYSIS.  —  S  i  n  c  e 

the  common  difference 
Com.  diff.  ,  2  X  6    :  is  2>  the  second  term  is 

First  term,  3  -j-  12  =  15,  the  7th  term.      equal  to  the  first  plus 

once  the  common  differ- 

ence, the  third  term  is  equal  to  the  first  plus  twice  the  common  differ- 
ence, the  fourth  term  is  equal  to  the  first  term  plus  three  times  the  com- 
mon difference.  Hence,  the  seventh  term  will  be  equal  to  the  first  term 
plus  six  times  the  common  difference,  which  is  15. 


.—J.ra/  term  of  an  arithmetical  progression  is  equal  to 
the  first  term,  increased  by  the  common  difference  multiplied  by  a 
number  one  less  than  the  number  of  terms. 
22 


338  PROGKESSIONS. 

2.  The  first  term  is  10  and  the  common  difference  5.    What 
is  the  10th  term?     Prove  it. 

3.  The  first  term  is   6   and   the   common  difference  is  8. 
What  is  the  25th  term  ? 

4.  A  boy  agreed  to  work  for  50  days  at  25  cents  the  first 
day,  and  an  increase  of  3  cents  per  day.    What  were  his  wages 
the  last  day  ? 

5.  A  body  falls  ISy1^  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  far  the  third  second.     How 
far  will  it  fall  the  seventh  second  ? 

6.  An  arithmetical  series  has  1000  terms,  the  first  term  of 
which  is  75  and  the  common  difference  5.     What  is  the  last 
term? 

7.  Find  the  sum  of  an  arithmetical  series  of  which  the  first 
term  is  2,  the  common  difference  3,  and  the  number  of  terms  7. 

PROCESS  ANALYSIS. — By  examining  the  series  2, 

2  4.  (  6  y  3  ^  =  2  0       5'  8>  n' 14'  17'  2°'  h  is  evident  that  the 

average  term  is  11,  for  if  half  the  sum 

2  -\-  20    —  2  2         of  any  two   terms  equidistant  from  the 

2  2  -r-    2  =  11         extremes  be  found  it  will  be  11,  and  in 

1  1  X     7=77         general  in  any  arithmetical  progression 

the  average  term  is  equal  half  the  sum 

of  the  extremes  or  any  two  terms  equidistant  from  the  extremes. 
Since  the  first  term  is  2  and  the  common  difference  3,  the  last  term 
is  found  by  the  previous  rule  to  be  20.  The  sum  of  the  extremes  is 
therefore  22,  which,  divided  by  2,  gives  the  average  term.  And  since 
there  are  7  terms,  the  sum  will  be  7  times  the  average  term,  or  77. 

RULE. — To  find  the  sum  of  an  arithmetical  series  :  Multiply 
half  the  sum  of  the  extremes  by  the  number  of  terms. 

8.  What  is  the  sum  of  an  arithmetical  series  composed  of 
50  terms,  of  which  the  first  term  is  2  and  the  common  differ- 
ence 3  ? 

9.  What  is  the  sum  of  a  series  in  which  the  first  term  is 
Y1^  the  common  difference,  y1^-,  and  the  number  of  terms  100? 


GEOMETKICAL   PROGRESSION.  339 

10.  A  man  walked  15  miles  the  first  day,  and  increased  his 
rate  3  miles  per  day   for  10  days.     How  far  did  he  walk  in 
the  eleven  days? 

11.  How  many  strokes  does  a  clock  strike  in  12  hours? 

12.  A  person  had  a  gift  of  $100  per  year  from  his  birth 
until  he  became  21   years  old.     These  sums  were  deposited 
in  a  bank  and  drew  simple  interest  at  6^.     How  much  was 
due  him  when  he  became  of  age? 


GEOMETRICAL  PROGRESSION. 

543.  A  Geometrical  Progression  is  a  series  of 
numbers  which  increase  or  decrease  by  a  constant  multiplier 
or  ratio. 

Thus,  5,  10,  20,  40,  80;  etc.,  is  a  geometrical  progression,  of  which 
the  multiplier  or  ratio  is  2. 

WRITTEN    EXERCISES. 

544.  1.  The  first  term  of  a  geometrical  series  is  3  and  the 
multiplier  or  ratio  is  2.     What  is  the  5th  term? 

PROCESS.  ANALYSIS. — Since   the   multiplier   is   2,  the 

2*=  1  6  second  term  will  be  3  X  .2,  the  third  3X2X2 

o  v  i  *  _  4  o        or  3  X  2L>,  the  fourth  3  X  22  X  2  or  3  X  23  and 

the  fifth  3  X  23  X  2  or  3  X  24,  that  is,  the  fifth 

term  is  equal  to  the  first  term  multiplied  by  the  ratio  raised  to  the 
fourth  power. 

RULE. — Any  term  of  a  geometrical  progression  is  equal  to  the 
first  term,  multiplied  by  the  ratio  raised  to  a  power  one  less  than 
the  number  of  the  term. 

2.  The  first  term  of  a  geometrical  progression  is  10,  the 
ratio  3.  What  is  the  6th  term? 


340  PROGRESSIONS. 

3.  The  first  term  of  a  geometrical  progression  is  10,  the 
ratio  4,  and  the  number  of  terms  6.     What  is  the  6th  term? 

4.  If  a  farmer  should  hire  a  man  for  10  days,  giving  him 
5  cents  for  the  first  day,  3  times  that  sum  for  the  second 
day,  and  so  on,  what  would  be  his  wages  for  the  last  day? 

5.  If  the  first  term  is  $100  and  the  ratio  1.06,  what  is 
the  6th  term  ?     Or,  what  is  the  amount  of  $100  at  compound 
interest  for  5  years  at  6%  ? 

6.  What  is  the  amount  of  $520  for  6  years,  at  5^  com- 
pound interest? 

7.  What  is  the  sum  of  a  geometrical  series,  of  which  the 
first  term  is  5,  the  ratio  3,  and  the  number  of  terms  5  ? 

PROCESS.  ANALYSIS.  —  Since     in 

this  series  the  first  term  is 
5  X  8  1  =  4  0  5,  the  5th  term.          ^  ^  ^  s>  and  the 

3X405  _  5  number  of  terms  5,  their 

3_1  =605,  the  sum.  Bum    may    be    obtained 

by  the  following  process, 
which  illustrates  the  formation  of  the  rule  : 

Series  5  +  15  +  45  +  135  +  405 
3  times  Series  15  +  45  +  135  +  405  +  1215 

2  times  Series        =  1215  —  5 

Series        = 


RULE.  —  The  sum  of  a  geometrical  series  is  equal  to  the  differ- 
ence between  the  first  term9  and  the  product  of  the  last  term  by  the 
ratio,  divided  by  the  difference  between  the  ratio  and  1. 

8.  The  extremes  of  a  geometrical  progression  are  4  and 
1024,  and  the  ratio  4.     What  is  the  sum  of  the  series? 

9.  The  extremes  are  ^  and  fff  and  the  ratio  2-|-.     What 
is  the  sum  of  the  series? 

10.  What  is  the  sum  of  the  series  in  which  the  first  term 
is  2,  the  last  term  0,  and  the  ratio  -^;  or  what  is  the  sum  of 
the  infinite  series  2,  1,  ^,  ^,  £,  ^  -^-,  etc? 


l7.laKEiU;ftliPKi 


DEFINITIONS. 

545.  Mensuration  treats  of  the  measurement  of  lines, 
surfaces,  and  solids. 

546.  A  Line  is  that  which  has  length  only. 

547.  A  Straight  Line  is  a  line  that 
does  not  change  its  direction. 

548.  A  Curved  Line  is  a  line  that 
changes  its  direction  at  every  point. 

549.  Parallel  Lines  are  such  as  are* 
equidistant  throughout  their  whole  extent. 


Straight  Line. 


Curved  Lines. 


Parallel  Lines. 


550.  A  Plane  Surface  is  a  surface  such  that  a  straight 
line  joining  any  two  points  of  it  is  wholly  in  the  surface. 

551.  A  Curved  Surface  is  a  surface  such  that  no  part 
of  it  is  a  plane  surface. 

552.  An  Angle  is   the  divergence  of 
two  lines  that  meet. 


553.  A  Right  Angle  is  the  angle 
formed  when  one  straight  line  meets  another 
making  the  adjacent  angles  equal. 

The  lines  are  perpendicular  to  each  other 
when  a  right  angle  is  formed. 


Angle. 


Two  Right  Angles. 
(341) 


342 


MENSURATION. 


Acute  Angle. 


Obtuse  Anglo. 


Triangle. 


Kec  tangle. 


Circle. 


554.  An  Acute  Angle  is  an    angle 
which  is  less  than  a  right  angle. 

555.  An  Obtuse  Angle  is  an  angle 
which  is  greater  than  a  right  angle. 

556.  The   Vertex  of  an  angle  is  the 
point  where  the  sides  meet. 

557.  A  Triangle  is  a  figure  which  has 
three  sides  and  three  angles. 

558.  A    Quadrilateral  is  a  figure 
bounded  by  four  sides. 

559.  A  Parallelogram,  is  a  quadri- 
lateral whose  opposite  sides  are  parallel. 

560.  A  Rectangle  is  a  parallelogram 
whose  angles  are  right  angles. 

561.  A    Polygon   is    a    plane   figure 
bounded  by  straight  lines. 

562.  A  Circle  is  a  plane  figure  bounded 
by  a   curved  line  every  point  of  which  is 
equally  distant  from  a  point  within  called 
the  center. 

563.  The  Circumference  is  the  line 
which  bounds  the  circle. 

564.  A  Hadius  of  a  circle  is  a  straight 
line  drawn  from  the  center  to  the  circum- 
ference. 

565.  A  Diameter  of  a   circle   is   a 
straight  line  drawn  through  the  center  and 
terminating  at  both  ends  in  the  circumfer- 
ence. 


MENSURATION. 


343 


Base. 


566.  The  Hase  of  a  figure  is  the  side 
on  which  it  is  assumed  to  stand. 

567.  The  Altitude  of  a  figure  is  the 
perpendicular  distance  between  the  base  and 
the  highest  point  opposite  it. 

56S.  A  Diagonal  of  a  figure  is  a 
straight  line  joining  the  vertices  of  two 
angles  not  adjacent. 

569.  The  Perimeter  of  a  figure  is  the  length  of  the 
lines  that  bound  it. 

570.  The  Area  of  a  surface  is  the  definite  amount  of 
surface  it  contains. 

MEASUREMENT  OF  LINES. 

571.  It  can  be  shown  by  geometry  that  the  circumference 
of  a  circle  is  3.1416  -f-  times  its  diameter. 

For  ordinary  measurements  it  is  sufficiently  accurate  to  consider 
the  circumference  3f  times  the  diameter. 

RULE. — 1.   The  circumference  is  equal  to  the  diameter  multi- 
plied by  3.1416. 

2.   The  circumference  divided  by  3. 1416  is  equal  to  the  diameter. 


WRITTEN    EXERCISES. 

572.    1.  What  is  the  circumference  of  a  circle  10  feet  in 
diameter? 

2.  What  is  the  circumference  of  a  circle  45  feet  in  diameter  ? 

3.  How  far  is  it  around  a  circular  lake  that  is  300  rods  in 
diameter  ? 

4.  What  is  the  circumference  of  a  circle  whose  radius  is  20 
rods? 


344  MENSURATION. 

5.  What  is  the  circumference  of  a  circle  whose  radius  is  5 
feet  6  inches  ? 

6.  What  is  the  diameter  of  a  circle  whose  circumference 
is  318.5  rods? 

7.  What  is  the  radius  of  a  circle  whose  circumference  ia 
1284  rods? 


MEASUREMENT  OF  SURFACES.     . 

573.  To  compute  the  area  of  a  parallelogram. 

The  truth  of  the  following  principle  has  been  shown  already : 

PRINCIPLE. — The  area  of  any  rectangular  figure  is  equal  to 
the  product  of  its  length  by  its  breadth  or  altitude. 


By  examining  the  figure  A,  B,  C,  D,  it  will 
be  seen  that  it  is  equal  to  E,  F,  D,  C,  and  that 
any  oblique  parallelogram  is  equal  to  a  rectangu- 
lar  parallelogram  of  the  same  base  and  altitude. 
Therefore, 

RULE. — The  area  of  any  parallelogram  is  equal  to  the  prod- 
uct of  the  base  multiplied  by  the  altitude. 


WRITTEN    EXERCISES. 

574.    1.  How  many  square   feet  are  there  in  a  parallelo- 
gram, whose  length  is  40  feet  and  altitude  13  feet? 

2.  What  is  the  area  of  a  parallelogram  whose  base  meas- 
ures 7  feet  and  whose  altitude  is  3  feet  8  inches  ? 

3.  What  is  the  area  of  a  field  in  the  form  of  a  parallelo- 
gram, whose  length  is  30  rods  and  the  perpendicular  distance 
between  the  sides  is  24  rods? 

4.  What  is   the  area  of  a  parallelogram  whose  length  is 
35  feet  and  whose  altitude  is  15  feet? 


MENSURATION.  345 

575.  To  compute  the  area  or  a  triangle. 

If  C  E  be  drawn  parallel  to  the  base  of  the 
triangle,  and  B  E  be  drawn  parallel  to  A  C, 
the  parallelogram  A  B  E  C  will  be  formed, 
of  which  the  original  triangle  is  one-half.  In 
the  same  manner  it  can  be  shown  that  every 

triangle  is  one-half  of  a  parallelogram  of  the  same  base  and  altitude. 
Therefore, 

RULE. — The  area  of  a  triangle  is  equal  to  one-half  the  prod- 
uct of  the  base  by  the  altitude. 

When  the  three  sides  are  given,  the  following  is  the  rule: 

RULE. — From  half  the  sum  of  the  three  sides  subtract  each  side 
separately.  Multiply  together  the  half  sum  and  the  three  remain- 
ders, and  extract  the  square  root  of  the  product.  The  result  will 
be  the  area  of  the  triangle. 

WRITTEN    EXERCISES. 

576.  1.  What  is  the  area  of  a    triangle  whose  base  is  24 
feet,  and  whose  altitude  is  18  feet? 

2.  What  is  the  area  of  a  triangle  whose  base  is  21  feet 
and  whose  altitude  is  12  feet? 

3.  What  is  the  cost,  at  $850  per  acre,  of  a  triangular  piece 
of  ground,  the  three  sides  of  which  are  in  a  ratio  of  5,  6 
and  8,  and  whose  shortest  side  is  120  feet? 

4.  What  is  the  area  of  a  triangle,  the  three  sides  of  which 
are  respectively  180  feet,  150  feet,  and  80  feet? 

5.  A  house  is  32  feet  wide,  and  the  rafters  are  20  feet 
long  on  each  side,  exclusive  of  any  projections.     What  will 
the  lumber  cost  at  $22.50  per  M,  which  will  inclose  both 
gable  ends  of  the  house? 

6.  What  is  the  area  of  a  triangle  whose  base  is  300  feet, 
and  whose  altitude  is  100  feet? 


346 


MENSURATION. 


577.   To  compute  the  area  of  a  polygon. 

Since  any  figure  may  be  divided  into  tri- 
angles, its  area  will  be  the  area  of  the  triangles 
which  compose  it.  Therefore, 

RULES. — I.  The  area  of  a  trapezium  is 
equal  to  the  diagonal,  multiplied  by  half  the 
sum  of  the  perpendiculars  drawn  from  the 
vertices  of  the  opposite  angles  to  the  diagonal. 

II.  Tlie  area  of  a  trapezoid  is  equal  to 
the  sum  of  the  parallel  sides  multiplied  by 
half  the  altitude. 

III.  The  area  of  a  regular  polygon  is 
equal  to  the  perimeter  of  the  polygon  mul- 
tiplied by   one -half  the  perpendicular  dis- 
tance from  the  center  to  one  of  the  sides  of 

^ie  polygon. 


WRITTEN    EXERCISES. 

578.  1.  What  is  the  area  of  a  trapezium,  the  diagonal  of 
which  is  110  feet,  and  the  perpendiculars  to  the  diagonal  are 
40  feet  and  60  feet  respectively? 

2.  The  parallel  sides  of  a  trapezoid  are  respectively,  10 
rods  and  8  rods,  and  the  altitude  6  rods.     What  is  its  area  ? 

3.  A  regular  octagon  has  a  perimeter  of  96  ft. ;  the  perpen- 
dicular from  the  center  to  one  side  is  12  ft.    What  is  its  area  ? 

4.  What  is  the  cost,  at  $125  per  acre,  of  a  piece  of  ground 
in  the  form  of  a  trapezoid,  whose  parallel  sides  are  respect- 
ively, 40  rods  and  30  rods,  and  whose  altitude  is  20  rods? 

5.  I  paid  $110  per  acre  for  a  piece  of  ground  in  the  form 
of  a -trapezium.     A  diagonal  line  crossing  it  was  120  rods 
long,  and  the   perpendiculars  drawn   to  the  diagonal  were, 
respectively,  30  rods  and  20  rods.     What  did  it  cost  me  ? 


MENSURATION. 


347 


579.  To  compute  the  area  of  a  circle. 

From  the  accompanying  figure  it  is 
evident  that  a  circle  may  be  regarded 
as  composed  of  a  large  number  of  tri- 
angles, the  sum  of  whose  bases  forms 
the  circumference  of  the  circle,  and 
whose  altitude  is  the  radius  of  the 
circle.  Hence, 

RULE. — 1.    The  area  of  a  circle  is  equal  to  the  circumference, 
multiplied  by  half  the  radius;  or, 

2.  — Multiply  the  square  of  the  diameter  by  ,7854, 


WRITTEN    EXERCISES. 

580,    1.  What  is  the  area  of  a  circle  whose  diameter  is 
5  feet? 

2.  What  is  the  area  of  a  circle  whose  diameter  is  8  feet  ? 

3.  What  is  the  area  of  a  circle  whose  circumference  is 
120  rods? 

4.  What  is  the  area  of  a  circle  whose  circumference  is  100 
feet? 

5.  A  gentleman  discovered  that  the  distance  around  a  cir- 
cular pond  was  320  rods.     What  was  its  area  ? 

6.  If  a  horse  is  tethered  to  a  stake  by  a  rope  15  rods  long, 
over  how  much  surface  can  he  graze? 

7.  How  long  must  a  rope  be  that  a  horse  can  graze  on 
just  an  acre? 

8.  The  area  of  a  circle  is  113.0976  square  rods.     What  is 
its  diameter? 

9.  The  round-house  of  the  P.  and  S.  Railroad  is  350  feet 
in  diameter.     How  much  land  does  it  cover? 

10.  What  is  the  area  of  a  railroad  turn-table  35  feet  in 
diameter? 


348  MENSURATION. 


MEASUREMENT  OF  SOLIDS. 

DEFINITIONS. 

581.  A  Solid  or  Body  is  that  which  has  length,  breadth 
and  thickness. 

The  planes  which  bound  a  solid  are  called  its  /aces,  and  their  inter- 
sections its  edges. 


Triangular          Quadrangular  Parallelopipcdon.  Cylinder. 

Prism.  Prism. 

582.  A  Prism  is  a  solid,  having  its  two  ends  equal  poly- 
gons, parallel  to  each  other,  and  its  sides  parallelograms. 

Prisms  are  named  from  the  form  of  their  bases  triangular, 
quadrangular,  pentagonal,  etc. 

583.  A  Parallelopipedon  is  a  solid  whose  opposite 
faces  are  equal  and  parallel  parallelograms. 

584.  A   Cylinder   is   a    regular    solid   bounded   by  a 
uniformly  curved  surface,  and  having  for  its  ends  two  equal 
circles,  parallel  to  each  other. 

The  face  of  any  section  of  a  cylinder  parallel  to  the  base  is  a  circle 
equal  to  the  base. 

585.  A  Pyramid  is  a  solid  whose  base  is  a  polygon 
and  whose  faces  are  triangles,  meeting  at  a  point  called  the 
vertex  of  the  pyramid. 


MEASUREMENT    OF   SOLIDS. 


349 


586.  A  Cone  is  a  solid,  whose  base  is  a  circle  and  whose 
surface  tapers  uniformly  to  a  point  called  the  vertex. 


Pyramid. 


Frustum  of  Cone. 


Frustum  of  Pyramid. 


587.  A  Frustum  of  a  pyramid  or  cone  is  the  portion 
remaining,  after  the  top  has  been  cut  off  by  a  plane  parallel 
to  the  base. 

588.  A  Sphere  is  a  solid,  every  point  of  whose  surface 
is  equally  distant  from  a  point  within,  called 

the  center. 

589.  The  Diameter  of  a  sphere  is  a 
straight  line  passing  through  the  center,  and 
terminating  in  the  surface  at  both  ends. 

Sphere. 

590.  The  Radius  of  a  sphere  is  one-half  the  diameter, 
or  the  distance  from. the  center  to  the  surface. 

591.  The    Circumference  of  a  sphere  is  the  greatest 
distance  around  the  sphere. 

592.  The  Altitude  of  a  solid  is  the  perpendicular  dis- 
tance from  its  highest  point  to  the  plane  of  the  base. 


CONVEX  SURFACE  OF  SOLIDS. 

593.  The  Convex  Surface  of  a  solid,  is  all  its  surface 
except  its  base  or  bases.  The  entire  convex  surface  includes 
the  area  of  the  bases  also. 


350 


MENSURATION. 


594.  To  find  the  convex  surface  of  a  prism  or  cylinder. 

It  is  evident  that  if  a  prism  or  cylinder  were  1  inch 
high,  its  convex  surface  would  contain  as  many  square 
units  of  surface  as  there  were  units  in  the  perimeter  of 
the  base;  and  if  it  were  2  inches,  3  inches,  or  4  inches 
high,  the  convex  surface  would  contain  2,  3,  or  4  times 
the  number  of  units  in  the  perimeter  of  the  base. 
Hence  the  following 

RULE. — Multiply  the  perimeter  of  the  base  by  the  altitude. 


WRITTEN    EXERCISES. 

595.  1.  What  is  the  convex  surface  of  a  cylinder  whose 
diameter  is  2  feet  and  length  5  feet? 

2.  What  is  the  convex  surface  of  a  quadrangular  prism 
whose  sides  are  each  2^  feet  and  whose  height  is  4  feet  ? 

3.  What  is  the  convex  surface  of  a  triangular  prism  whose 
sides  are  each  6  feet  and  whose  altitude  is  8  feet? 

4.  What  is  the  entire  surface  of  a  cylinder  which  is  5  feet 
in  length,  and  whose  base  is  2  feet  in  diameter? 

5.  What  is  the  convex  surface  of  a  piece  of  timber  in  the 
form  of  a  triangular  prism,  which  is  18  feet  long  and  the  sides 
of  whose  base  are  10  inches,  14  inches,  and  18  inches? 

596.  To  lifiicl  the  convex  surface  of  a  pyramid  or  cone. 

It  is  evident  that  the  convex  surface  of  any  pyra- 
mid is  composed  of  triangles,  and  the  convex  surface 
of  a  cone  may  also  be  assumed  to  be  made  up  of  an 
infinite  number  of  triangles.  The  bases  of  these 
triangles  form  the  perimeter  of  the  solid,  and  their 
height  is  the  slant  height  of  the  solid.  Therefore 
the  following  is  the  rule: 

RULE. — Multiply  the  perimeter  of  the  base  by  one-half  the  slant 
height. 


MEASUREMENT   OF   SOLIDS.  351 


WRITTEN     EXERCISES. 

597.  1.  What  is   the  convex   surface  of  a  quadrangular 
pyramid,  whose  base  is  15  feet  square  and  the  slant  height  18 
feet? 

2.  What  is  the  convex  surface  of  a  cone  whose  diameter  at 
the  base  is  12  feet  and  whose  slant  height  is  20  feet? 

3.  What  is  the  convex  surface  of  a  cone  whose  base  is  20 
feet  in  diameter  and  whose  slant  height  is  20  feet? 

4.  What  is  the  cost  of  painting  a  church  steeple,  the  base 
of  which  is  an  octagon  6  feet  on  each  side,  and  whose  slant 
height  is  80  feet,  at  $.30  per  square  yard? 

5.  How  many  feet  of  convex  surface  are  there  on  a  cone,  the 
base  diameter  of  which  is  6  feet  and  whose  slant  height  is  9-J- 
feet? 

6.  How  many  feet  of  convex  surface  are  there  on  a  pyra- 
mid whose  base  is  10  feet  square  and  whose  slant  height  is 
20  feet? 

7.  How  many  feet  of  convex  surface  are  there  on  a  cone 
whose  base  is  8  feet  in  diameter  and  whose  slant  height  is 
6  feet? 

8.  What  is  the  convex  surface  of  a  cone  whose  base  is  10 
feet  in  diameter  and  whose  slant  height  is  10  feet? 

598.  To  find  the  convex  surface  of  a  frustum  of  a 
pyramid  or  cone. 

It  is  evident  that  the  convex  surface  of  a  frustum 
of  a  pyramid  is  composed  of  trapezoids,  the  sum  of 
whose  parallel  sides  forms  the  perimeter  of  the  bases, 
and  whose  altitude  is  the  slant  height  of  the  frustum; 
and  the  convex  surface  of  a  cone  may  be  assumed  to  be  made  of  an 
infinite  number  of  trapezoids.  Hence, 

RULE. — Multiply  half  the  sum  of  the  perimeter  of  the  two  bases 
by  the  slant  height. 


352  MENSUEATION. 


WRITTEN     EXERCISES. 

599.  1.  How  many  feet  of  convex  surface  are  there  in  the 
frustum  of  a  cone  whose  slant  height  is  8  feet,  the  diameter 
of  whose  lower  base  is  12  feet  and  upper  base  8  feet? 

2.  What  is  the  convex  surface  of  the  frustum  of  a  pyramid 
the  slant  height  of  which  is  25  feet,  whose  lower  base  is  40  feet 
square,  and  whose  upper  base  is  20  feet  square  ? 

3.  What  did  it  cost,  at  $  .15  per  sq.  yd.,  to  paint  the  con- 
vex surface  of  a  vat  which  was  10  feet  in  diameter  at  the 
bottom  and  8  feet  at  the  top,  the  slant  height  of  which  was 
12  feet? 

4.  What  is  the  convex  surface  of  a  vat,  the  base  of  which 
is  9  feet  square  whose  top  is  8  feet  square  and  whose  slant 
height  is  10  feet? 

5.  What  would  the  lumber  cost  at  $40  per  M,    to  build 
such  a  vat  if  the  sides  were  of  1|-  inch  plank,  and  the  bottom 
was  2  inch  plank  ? 

600.  To  find  the  convex  surface  of  a  sphere. 

The  convex  surface  of  a  sphere  is  computed,  according  to  geomet- 
rical principles,  by  the  following  rule: 

EULE. — 1.  Multiply  the  diameter  by  the  circumference. 
2.  Multiply  the  square  of  the  diameter  by  3.1416. 

EXER  CISES. 

601.  1.  What  is  the  convex  surface  of  a  sphere  whose 
diameter  is  15  inches? 

2.  What  is  the  convex  surface  of  a  spherical  cannon-ball  8 
inches  in  diameter? 

3.  What  is  the  convex  surface  of  a  base-ball  whose  circum- 
ference is  9|-  inches? 


MEASUREMENT    OF   SOLIDS.  353 

4.  What  is  the  convex  surface  of  a  sphere  whose  circum- 
ference is  12  feet? 


VOLUME  OF  SOLIDS. 

602.  The  Volume  of  any  body  is  the  number  of  solid 
units  it  contains. 

'  603.  To  find  the  volume  of  a  prism  or  cylinder. 

It  is  evident  that  if  a  prism  or  cylinder  were  1  inch 
high,  it  would  contain  as  many  cubic  inches  as  there 
were  square  inches  in  the  area  of  the  base;  and  if  it 
were  2  inches,  3  inches,  or  4  inches  high,  the  volume 
would  be  2  or  3  or  4  times  as  much.  Hence  the  fol- 
lowing is  the  rule: 

RULE. — Multiply  the  area  of  the  base  by  the 
altitude. 

WRITTEN    EXERCISES. 

604.    1.  What  are  the  solid  contents  of  a  prism    whose 
base  is  12  inches  square  and  whose  height  is  2  feet  ? 

2.  What  is  the  volume  of  a  cylinder  whose  diameter  is  1|- 
feet  and  whose  length  is  4  feet? 

3.  What  would  be  the  cost  of  a  piece  of  timber  20  feet  long, 
18  inches  wide  and  12  inches  thick  at  $.30  per  cubic  foot? 

4.  What  will  be  the  capacity  in  bushels  of  a  square  bin  the 
base  of  which  was  8  feet  square,  and  the  height  of  which  was 
9  feet  on  the  inside  ? 

5.  How  many  gallons  of  water  will  a  vat  in  the  form  of  a 
cylinder  hold,  whose  inside  dimensions  are — base  8  feet  in 
diameter,  height  7  feet? 

6.  How  much  would  the  wheat  be  worth  at  $1.85  per 
bushel,  which  would  just  fill  a  bin  the  base  of  which  is  15 
feet  square,  and  the  height  of  which  is  12  feet? 

23 


354  MENSURATION. 

605.  To  find  the  volume  of  a  pyramid  or  cone. 

It  can  be  shown  by  geometry  that  a  pyramid  or  cone  is  one-third 
of  a  prism  or  cylinder  of  the  same  base  and  altitude.  Hence  the  fol- 
lowing is  the 

RULE.—  Multiply  the  area  of  the  base  by  one-third  of  the  altitude. 


WRITTEN    EXERCISES. 

606.  1.  What  are  the  solid  contents  of  a  cone,  the  diam- 
eter of  whose  base  is  6  feet  and  whose  altitude  is  9  feet  ? 

2.  What  are  the  solid  contents  of  a  pyramid  whose  base  is 
30  feet  square  and  whose  altitude  is  60  feet? 

3.  If  a  cubic  foot  of  granite  weighs  165  lb.,  what  is  the 
weight  of  a  granite  cone  the  diameter  of  whose  base  is  6  feet 
and  whose  altitude  is  8  feet  ? 

4.  What  is  the  weight  of  a  marble  pyramid  whose  base  is 
4  feet  square  and  whose  altitude  is   8   feet,  if  a  cubic  foot 
of  marble  weighs  171  pounds  ? 

607.  To  find  the  volume  of  a  frustum  of  a  pyramid 
or  cone. 

It  can  be  shown  by  geometry  that  the  frustum  of  a  pyramid  or 
cone  is  equal  to  three  pyramids  or  cones,  having  for  their  bases,  re- 
spectively, the  upper  base  of  the  frustum,  its  lower  base,  and  a  mean 
proportional  between  the  two  bases.  Hence  the  following  is  the 

RULE. — To  the  sum  of  the  areas  of  the  two  ends  add  the  square 
root  of  the  product  of  these  areas,  and  multiply  the  result  by  one- 
third  of  the  altitude. 

EXER  CISES. 

608.  1.  What  is  the  volume  of  a  frustum  of  a  pyramid 
the  lower  base  of  which  is  20  feet  square,  the  upper  base  10 
feet  square  and  the  altitude  20  feet? 


MEASUREMENT   OF   SOLIDS.  355 

2.  What  are  the  solid  contents  of  the  frustum  of  a  cone 
whose  upper  base  is  5  feet  in  diameter,  whose  lower  base  is 
8  feet  in  diameter,  and  whose  altitude  is  7  feet? 

3.  A  tree  was  3  feet  in  diameter  at  the  butt  and  its  diam- 
eter at  a  height  of  40  feet  was  1  foot.     What  were  the 
cubical  contents  of  that  portion  of  the  tree? 

4.  A  vat  whose  inside  measurements  were  as  follows — 
diameter  of  the  bottom  12  feet,  diameter  of  the  top  10  feet, 
height  9  feet — was  filled  with  water.     How  many  gallons 
did  it  contain? 

609.  To  find  the  volume  or  contents  of  a  sphere. 

A  sphere  may  be  regarded  as  composed  of  pyra- 
mids whose  bases  form  the  surface  of  the  sphere, 
and  whose  altitude  is  the  radius  of  the  sphere. 
Hence  the  following  is  the 

KULE. — 1.  Multiply  the  convex  surface  by 
one -third  of  the  radius;  or, 

2.  Multiply  the  cube  of  the  diameter  by  .5236. 


EXERCISES. 

610.    1.  The  diameter  of  a  sphere  is  5  feet.     How  many 
cubic  feet  does  it  contain? 

2.  Find  the  contents  of  a  sphere  whose  diameter  is  8  feet. 

3.  The  circumference  of  a  sphere  is  9.4248.     What  are  its 
cubical  contents? 

4.  A  cubic  foot  of  cast-iron  weighs  about  450  pounds. 
What  is  the  weight  of  a  cannon-ball  whose  diameter  is  18 
inches? 

5.  What  are  the  cubical  contents  of  a  spherical  vessel  the 
diameter  of  which  is  2|-  feet? 

6.  How  many  cubic  feet  are  there  in  a  spherical  body  whose 
diameter  is  25  feet  ? 


356  MISCELLANEOUS   EXAMPLES. 


MISCELLANEOUS  EXAMPLES. 

611.    1.  If  5  men  can  do  a  piece  of  work  in  12  days,  how 
long  will  it  take  6  men  to  do  the  same  work? 

2.  If  5  barrels  of  apples  cost  $7.50,  what  will  8  barrels 
cost  at  the  same  rate? 

3.  It  required  20  men  to  load  a  vessel  in  6  days,  how 
many  men  would  it  require  to  load  it  in  1^-  days  ? 

4.  A  steamboat  sailed  42|-  miles  in  2|-  hours.     How  far 
did  she  sail  in  20  minutes? 

5.  If  6  men  can  dig  28  rods  of  ditch  in  1  day,  how  many 
men  will  it  require  to  dig  56  rods  in  f  of  a  day? 

6.  If  |  of  a  yard  of  broadcloth  cost  $3f ,  what  will  £  of  a 
yard  cost? 

7.  If  it  costs  $50  to  support  a  family  of  8  persons  for  2^- 
weeks,  what  will  it  cost  to  support  10  persons  3  weeks? 

8.  If  3  pounds  of  tea  are  worth  14  pounds  of  coffee,  and 
5  pounds  of  coffee  are  worth  18   pounds  of  sugar,  and  21 
pounds  of  sugar  are  worth  60  pounds  of  flour,  how  many 
pounds  of  flour  are  equal  in  value  to  7  pounds  of  tea  ? 

9.  A  farmer  sold  12  firkins  of  butter,  each  containing  56 
pounds,  for  23  cents  a   pound,  and  received  in  payment  5 
pounds  tea  at  85  cents  per  pound,  60  pounds  sugar  at  13 
cents  per  pound,  15  yards  cloth  at  $1.121  per  yard,  and  the 
balance  in  money.     How  much  money  did  he  receive? 

10.  A  regiment  of  soldiers  consisting  of  1100  men,  was 
furnished  with  bread  sufficient  to  last  it  8  weeks,  allowing 
each  man  15  oz.  per  day.     If  ^  of  it  was  found  to  be  unfit 
for  use,  how  many  ounces  per  day  shall  each  man  receive  so 
that  the  balance  may  last  8  weeks? 

11.  A  man  being  asked  how  many  sheep  he  had,  replied, 
"  If  I  had  3  times  as  many  as  I  have  and  5  sheep,  I  would 
have  185."     How  many  had  he? 


MISCELLANEOUS    EXAMPLES.  357 

12.  A  man  paid  \  of  his  money  on  a  debt,  \  of  the  remain- 
der for  a  suit  of  clothes,  \  of  the  remainder  for  provisions, 
and  lost  \  of  the  remainder,  when  he  had  $5   left.     How 
much  had  he  at  first? 

13.  Three  men  engage  to  reap  a  field  of  wheat.     A  can 
do  it  in  15  days,  B  in  18  days  and  C  in  20  days.     In  what 
time  can  they  do  it  together  ? 

14.  A  farmer  was  offered  $1.45  per  bu.  for  his  wheat,  but 
determined  to  have  it  ground  and  sell  the  flour.     It  cost  to 
take  it  to  the  mill  2^  cents  per  bu. ;   the  miller  took  \  for 
grinding;  it  took  4|£  bu.  to  make  a  barrel  of  flour;  he  paid 
45  cents  apiece  for  barrels,  and  it  cost  25  cents  per  barrel 
commission  to  sell  it.     75  bbl.  sold  for  $550  and  25  bbl.  for 
$165.     If  the  refuse  was  sold  for  $100,  did  he  make  or  lose, 
and  how  much  per  hundred  barrels? 

15.  A  farmer  being  asked  how  many  apple-trees  he  had, 
replied,  "  If  I  had  3  times  as  many  and  5  trees  more,  I  would 
have  1358."     How  many  had  he? 

16.  \  of  A's  money  is  equal  to  •§  of  B's,  and  the  difference 
is  $8.     How  much  has  each  ? 

17.  A,  B  and  C  hire  a  pasture  for  $170.     A  puts  in  70 
sheep  for  6^  months,  B  24  cattle  for  4^-  months,  C  10  cattle 
and  35  sheep  for  5|-  months.     If  2  cattle  eat  as  much  as  7 
sheep,  how  much  should  each  pay? 

18.  If  a  pole  10  feet  long,  casts  a  shadow  13  feet  long, 
what  is  the  length  of  a  pole  which  will  cast  a  shadow  62^- 
feet  long  at  the  same  time? 

19.  A's  weight  is  f  that  of  B,  and  C's  weight  is  as  much 
as  A's  and  B's  together.     The  sum  of  their  weights  is  490 
pounds.     What  is  the  weight  of  each  ? 

20.  f  of  A's  money  is  equal  to  f  of  B's,  and  the  difference 
is  $5.     How  much  money  has  each  ? 

21.  The  ages  of  A,  B  and  C,  are  to  each  other  as  3,  4 
and  5,  and  their  sum  is  136  years.     What  is  the  age  of  each? 


358  MISCELLANEOUS   EXAMPLES. 

22.  A  boy  bought  a  certain  number  of  apples  at  the  rate 
of  4  for  5  cents,  and  sold  them  at  the  rate  of  3  for  4  cents. 
He  gained  60  cents.     How  many  did  he  buy  ? 

23.  A,  B  and  C  agree  to  build  a  house.     A  and  B  can  do 
the  work  in  32  days,  B  and  C  in  28  days,  and  A  and  C  in 
26  days.     How  long  will  it  take  them  to  do  it  by  working 
together?     How  long  would  it  take  each  to  do  it  alone? 

24.  A  can  build  a  wall  in  10  days,  by  working  12  hours 
a  day,  B  can  build  it  in  9  days,  by  working  10  hours  a  day. 
In  how  many  days  can  both  build  it,  by  working  8  hours  a 

.day? 

25.  A  pair  of  horses  is  sold  for  $390.     One  of  them  is 
worth  |-   as    much   as   the   other.     What  is    the   value   of 
each? 

26.  A  hind  wheel  of  a  carriage  4  feet  6  inches  high,  re- 
volved 720  times  in  going  a  certain  journey.     How  many 
revolutions  did  the  fore  wheel  make,  which  was  4  feet  high  ? 

27.  The  shadow  of  a  pole  6  feet  long  is  9  inches,  and  the 
shadow  of  a  steeple  at  the  same  time  is  9  feet  long.     What 
is  the  height  of  the  steeple  ? 

28.  What  is  the  bank  discount  on  a  note  for  $245.30,  due 
in  90  days,  if  discounted  at  6^  ? 

29.  If  a  man  takes  2  steps  of  30  inches  each  in  3  seconds, 
how  long  will  it  take  him  to  walk  10  miles? 

30.  It  cost  $150  to  support  4  grown  persons  and  3  children 
8  weeks.     What  will  it  cost  to  support  3  grown  persons  and 
8  children  for  the  same  time,  if  3  children  cost  as  much  as 
2  grown  persons? 

31.  A  man  bought  20  bushels  of  wheat  and  15  bushels  of 
corn  for  $36,  and  15  bushels  of  wheat  and  25  bushels  of  corn 
for  $32.50?     What  did  he  pay  per  bu.  for  each? 

32.  A  fox  has  120  rods  the  start  of  a  hound.    If  the  hound 
runs  30  rods  while  the  fox  runs  26,  how  far  will  the  hound 
run  before  he  overtakes  the  fox? 


MISCELLANEOUS    EXAMPLES.  359 

33.  A  starts  on  a  journey  at  the  rate  of  3  miles  an  hour. 
6  hours  afterward,  B  starts  after  him  at  the  rate  of  4  miles 
an  hour.     How  far  will  B  travel  before  he  overtakes  A? 

34.  One-fourth  of  a  certain  number  is  10  more  than  -J-  of 
it.     What  is  the  number? 

35.  If  to  a  certain  number  you  add  ^  of  itself  and  -^  of 
itself,  the  sum  will  be  105.     What  is  the  number? 

36.  If  to  a  certain  number  you  add  15  more  than  |  of 
itself,  the  sum  will  be  40.     What  is  the  number? 

37.  How  many  days  will  it  take  30  men  to  do  a  piece  of 
work,  which  20  men  can  do  in  45  days? 

38.  If  a  man  can  earn  -|  of  a  dollar  in  f  of  a  day,  how 
much  can  he  earn  in  f  of  a  day? 

39.  How  many  yards  of  silk  f  yard  wide,  will  it  take  to 
line  4^  yards  of  broadcloth  If  yards  wide? 

40.  If  14  ounces  of  wool  make  2J  yards  of  cloth  1  yard 
wide,  how  much  will  it  take  to  make  6|-  yards   1^  yards 
wide? 

41.  How  many  tiles  14  inches  long,  will  it  take  to  make 
a  drain  which  is  -|-  of  a  mile  long  ? 

42.  If  $300  placed  at  interest  yields  an  income  of  $18  in 
9  months,  how  much  must  be  placed  at  interest  at  the  same 
rate  to  yield  an  income  of  $115  in  6  months? 

43.  If  to  a  certain  number  you  add  \  of  itself,  the  result  will 
be  20  less  than  double  the  number.     What  is  the  number  ? 

44.  At  what  time  between  5  and  6  o'clock  will  the  hour 
and  minute  hands  of  a  clock  be  exactly  together  ? 

45.  Two  soldiers  start  together  for  a  certain  fort.     One, 
who  travels  12  miles  per  day,  after  traveling  9  days,  turns 
back  as  far  as  the  other  had  traveled  during  those  9  days. 
He  then  turns  and  pursues  his  way  toward  the  fort,  where 
both  arrive  together  18  days  from  the  time  they  set  out.     At 
what  rate  did  the  other  travel? 

46.  A  man  agreed  to  execute  a  piece  of  work  in  60  days, 


360  MISCELLANEOUS    EXAMPLES. 

and  employed  30  men  to  perform  the  labor.  At  the  end  of 
40  days  it  was  only  half  finished.  How  many  additional 
laborers  was  he  obliged  to  employ  to  perform  the  work  within 
the  time  agreed  upon? 

47.  A  person,  being  asked  the  time  of  day,  replied  that  it 
was  past  noon,  and  that  f  of  the  time  past  noon  was  equal  to 
f  of  the  time  to  midnight.     What  was  the  time  ? 

48.  A  gentleman  wishes  his  son  to  have  $3000  when  he 
is  21  years  of  age.     What  sum  must  be  deposited  at  the  son's 
birth,  in  a  savings  bank,  which  pays  compound  interest  at 
the  annual  rate  of  6^,  so  that  the  deposit  shall  amount  to 
that  sum  when  the  boy  becomes  of  age? 

49.  A  note  for  $100  was  due  on  Sept.  1st,  but  on  Aug.  llth 
the  maker  proposed  to  pay  as  much  in  advance  as  will  allow 
him  (?0  days  after  Sept.  1st  to  pay  the  balance.     How  much 
must  be  paid  Aug.  llth,  money  being  worth  6%? 

50.  What  sum  must  a  person  save  annually,  commencing 
at  21  years  of  age,  so  that  he  may  be  worth  $25000  when  he 
is  40  years  old,  if  he  gets  6^   compound  interest  on  his 
money? 

51.  If  a  merchant  sells  f  of  an  article  for  what  -J  of  it  cost, 
what  is  his  gain  per  cent.  ? 

52.  If  goods  are  sold  so  that  -f  of  the  cost  is  received  for 
half  the  quantity  of  goods,  what  is  the  gain  per  cent.  ? 

53.  A  man  sold  a  horse  and  carriage  for  $597,  gaining  by 
the  sale  25^  on  the  cost  of  the  horse  and  10^  on  the  cost  of 
the  carriage.     If  f  of  the  cost  of  the  horse  equaled  -|  of  the 
cost  of  the  carriage,  what  was  the  cost  of  each  ? 

54.  If  300  cats  can  kill  300  rats  in  300  minutes,  how  many 
cats  can  kill  100  rats  in  100  minutes? 

55.  A  party  of  8  hired  a  coach.     If  there  had  been  4  more 
the  expense  would  have  been  reduced  $1  for  each  person. 
How  much  was  paid  for  the  coach? 

56.  I  sold  goods  at  a  gain  of  20%.     If  they  had  cost  me 


MISCELLANEOUS    EXAMPLES.  361 

$250  more  than  they  did,  I  would  have  lost  20%  by  the  sale. 
How  much  did  the  goods  cost  me? 

57.  A  laborer  agreed  to  work  for  $1.25  per  day  and  his 
board,  paying  $  .50  per  day  for  his  board  when  he  was  idle. 
At  the  end  of  25  days  he  received  $19.     How  many  days 
was  he  idle? 

58.  A  is  20  years  of  age;  B's  age  is  equal  to  A's  and  half 
of  C's;   and  C's  is  equal  to  A's  and  B's  together.     What  is 
the  age  of  each? 

59.  A  and  B  were  partners  in  a  profitable  enterprise.     A 
put  in  $4500  capital  and  received  -f  of  the  profits.     What 
was  B's  capital? 

60.  A  man  spent  $4  more  than  half  his  money  traveling, 
one-half  what  he  had  left  and  $2  more  for  a  coat,  $6  more 
than  half  the  remainder  for  other  clothing,  and  had  $2  left. 
How  much  money  had  he  at  first? 

61.  A  boy  bought  at  one  time  5  apples  and  6  pears  for  28 
cents,  and  at  another  time  6  apples  and  3  pears  for  21  cents. 
What  was  the  cost  of  each  kind  of  fruit  ? 

62.  A  and  B  can  do  a  piece  of  work  in  20  days.     If  A 
does  f  as  much  as  B,  in  how  many  days  can  each  do  it  ? 

63.  A  man  bought  a  farm  for  $5000,  agreeing  to  pay  prin- 
cipal and  interest  in  5  equal  annual  installments.     What  will 
be  the  annual  payment,  including  interest  at  6%? 

64.  A  carriage  maker  sold  two  carriages  for  $300  each. 
On  one  he  gained  25^  ;   on  the  other  he  lost  25^.     Did  he 
gain  or  lose  by  the  sale?     How  much,  and  how  much  per 
cent.  ? 

65.  If  a  ladder  placed  8  feet  from  the  base  of  a  building 
40  feet  high,  just  reached  the  top,  how  far  must  it  be  placed 
from  the  base  of  the  building  that  it  may  reach  a  point  10 
feet  from  the  top? 

66.  Mr.  A.  is  35  years  of  age  and  his  son  is  10.     How 
soon  will  the  son  be  one-half  the  age  of  the  father? 


-    t 
362  MISCELLANEOUS    EXAMPLES. 

67.  A  person  in  purchasing  sugar  found  that  if  he  bought 
sugar  at  11  cents  he  would  lack  30  cents  of  having  money 
enough  to  pay  for  it,  so  he  bought  sugar  at  10^-  cents  and 
had  15  cents  left.     How  many  pounds  did  he  buy? 

68.  A  farmer  had  his  sheep  in  three  fields,     f  of  the  num- 
ber in  the  first  field  was  equal  to  f  of  the  number  in  the 
second  field,  and  -f  of  the  number  in  the  second  field  was  f  of 
the  number  in  the  third  field.     If  the  entire  number  was  434, 
how  many  were  there  in  each  field? 

69.  A  and  B  can  do  a  piece  of  work  in  10  days,  B  and  C 
can  do  it  in  12  days,  and  A  and  C  in  15  days.     How  long 
will  it  take  each  to    do  it? 

70.  A,  B  and  C  pasture  an  equal  number  of  cattle  upon  a 
field  of  which  A  and  B  are  the  owners — A  of  9  acres  and  B 
of  15  acres.     If  C  pays  $24  for  his  pasturage,  how  much 
should  A  and  B  each  receive? 

71.  How  many  acres  are  there  in  a  square  tract  of  land 
containing  as  many  acres  as  there  are  boards  in  the  fence 
inclosing  it,  if  the  boards  are  11  feet  long  and  the  fence  is 
4  boards  high? 

72.  What  is  the  greatest  number  which  will  divide  27,  48, 
90,  and  174,  and  leave  the  same  remainder  in  each  case? 

73.  A  and  B  invested  equal  sums  in  business.     A  gained 
a  sum  equal  to  25^  of  his  stock,  and  B  lost  $225.     A's 
money  at  this  time  was  double  that  of  B's.     What  amount 
did  each  invest? 

74.  A  man  at  his  marriage  agreed  that  if  at  his  death  he 
should  leave  only  a  daughter,  his  wife  should  have  f  of  his 
estate;  and  if  he  should  leave  only  a  son,  she  should  have  \. 
He  left  a  son  and  a  daughter.     What  fractional  part  of  the 
estate  should  each  receive,   and  how  much  was  each  one's 
portion,  if  the  estate  was  worth  $6591  ? 


TEST   QUESTIONS.  363 


TEST  QUESTIONS. 

612.  Define  a  unit;  a  number.  Explain  the  necessity  for  a  uni- 
form system  of  grouping  objects,,  In  how  many  ways  may  numbers 
be  represented?  Name  them.  Define  numeration;  notation;  Arabic 
notation;  Roman  notation.  Give  the  first  principle  of  Arabic  nota- 
tion. Illustrate  it.  What  is  meant  by  "units  of  first  order,"  etc.? 
Give  the  general  principles  of  Arabic  notation.  What  is  meant  by 
a  period  of  figures?  Give  the  names  of  the  first  seven  periods.  Give 
the  rule  for  notation  ;  for  numeration.  State  how  cents  and  mills 
are  written  in  notation  of  U.  S.  money.  What  characters  are  em- 
ployed in  Roman  notation?  Give  the  principles  of  Roman  notation. 

Define  addition;  sum,  or  amount;  equation;  like  numbers.  De- 
scribe the  sign  of  addition;  the  sign  of  equality.  How  many  cases 
are  there  in  addition?  Show  the  truth  of  the  principles  of  addition. 
Repeat  the  rule  for  addition.  Why  do  we  begin  at  the  right  to  add? 
Why  are  the  numbers  of  the  same  order  written  in  the  same  column? 

Define  subtraction;  minuend;  subtrahend;  remainder;  difference. 
What  is  the  sign  of  subtraction?  WThat  is  it  called?  State  the  prin- 
ciples of  subtraction.  Show  that  they  are  true.  Explain  what  is  to 
be  done  when  some  figure  of  the  subtrahend  expresses  more  than  the 
corresponding  figure  of  the  minuend. 

Define  multiplication;  multiplicand;  multiplier;  product;  factors 
of  the  multiplier ;  abstract  number.  Describe  the  sign  of  multipli- 
cation. Give  the  principles  of  multiplication.  Show  that  they  are 
true.  Show  that  multiplication  is  a  special  case  of  addition.  Repeat 
the  rule  for  multiplication.  What  steps  in  the  process  are  for  con- 
venience? How  may  you  multiply  when  there  are  ciphers  on  the 
right  of  either  or  both  factors? 

Define  division;  dividend;  divisor;  quotient;  remainder.  What  is 
the  sign  of  division?  In  how  many  ways  is  division  indicated? 
State  the  principles  of  division.  Show  that  they  are  true.  Show  that 
division  is  a  special  case  of  subtraction.  In  how  many  ways  may 
the  remainder  be  expressed?  Illustrate  each  way  by  an  example. 
WThat  is  a  fraction?  What  is  meant  by  long  division?  What  is 
meant  by  short  division?  Which  should  precede  the  other?  Why? 
What  steps  in  the  process  of  division  are  for  convenience?  What  are 
necessary?  How  may  you  proceed  when  there  are  ciphers  on  the 


364  TEST   QUESTIONS. 

right  of  either  divisor  or  dividend?  State  the  principles  governing 
the  relation  of  dividend,  divisor,  and  quotient.  Illustrate  each  by 
an  example.  Define  analysis.  Illustrate  the  process.  Describe  the 
parenthesis  and  vinculum,  and  show  their  uses. 

Define  and  illustrate  what  is  meant  by  an  integer;  exact  divisor; 
factor;  a  prime  number;  a  composite  number;  an  even  number;  an 
odd  number.  Give  eleven  facts  relating  to  exact  divisibility  of  num- 
bers. Illustrate  each  statement  by  an  appropriate  example.  What  is 
meant  by  factoring?  Prime  factors?  What  is  an  exponent?  State  the 
principles  relating  to  the  prime  factors  of  numbers.  Illustrate  the 
truth  of  these  principles  by  appropriate  examples.  Give  the  rule  for 
finding  the  prime  factors  of  a  number.  Explain  the  process  of  multi- 
plying by  factors.  Show  the  use  of  this  process.  Show  how  to  divide 
by  factors.  Explain  how  to  find  the  true  remainder  in  division  by 
factors.  Give  the  rule  for  dividing  by  the  factors  of  a  number. 

What  is  meant  by  cancellation?  Upon  what  principle  is  the  pro- 
cess based?  Illustrate  the  process. 

Define  what  is  meant  by  a  common  divisor;  the  greatest  common 
divisor;  numbers  that  are  prime  to  each  other.  What  is  the  princi- 
ple underlying  the  greatest  common  divisor?  Give  the  ordinary 
method  of  finding  the  greatest  common  divisor  when  the  numbers 
are  small.  Solve  an  example,  and  give  the  analysis  when  the  num- 
bers can  not  be  readily  factored. 

What  is  a  multiple?  Define  what  is  meant  by  a  common  multi- 
ple; the  least  common  multiple.  State  the  principle  upon  which  the 
processes  in  least  common  multiple  are  based.  Solve  an  example 
showing  the  truth  of  the  principle. 

Define  and  illustrate  what  is  meant  by  the  terms  fraction ;  unit  of  a 
fraction;  fractional  unit ;  the  denominator;  the  numerator;  the  terms  of 
a  fraction;  a  proper  fraction ;  an  improper  fraction;  a  mixed  number; 
a  common  fraction;  a  decimal  fraction.  How  are  fractional  expres- 
sions read?  Interpret  the  expression  f-. 

What  is  meant  by  reduction  of  fractions?  What  is  Case  I?  When 
is  a  fraction  reduced  to  larger  or  higher  terms?  Upon  what  princi- 
ple does  the  process  in  Case  I  depend?  What  is  Case  II?  What  is 
meant  by  reducing  a  fraction  to  smaller  or  lower  terms?  To  smallest 
or  lowest  terms?  Upon  what  principle  is  the  process  in  Case  II 
based?  What  is  Case  III  in  reduction?  Solve  an  example  illustrat- 
ing the  process.  What  is  Case  IV?  Solve  an  example  illustrating 


TEST   QUESTIONS.  365 

the  process.  What  is  meant  by  similar  fractions?  Dissimilar  frac- 
tions? When  have  similar  fractions  their  least  common  denominator? 
Give  the  principles  relating  to  the  common  and  least  common  denom- 
inator of  fractions.  What  is  the  rule  for  finding  the  least  common 
denominator  of  several  fractions? 

What  kind  of  fractions  only  can  be  added?  Why?  What  must 
be  done  with  dissimilar  fractions  before  they  can  be  added?  How 
should  mixed  numbers  be  added?  What  kind  of  fractions  only  can 
be  subtracted?  What  must  be  done  to  dissimilar  fractions  before  they 
can  be  subtracted?  How  could  mixed  numbers  be  subtracted? 

What  is  Case  I  in  multiplication  of  fractions?  What  principle  un- 
derlies the  process?  Demonstrate  the  truth  of  the  principle.  What 
is  Case  II?  What  is  the  principle?  What  is  Case  III?  What  is  the 
general  rule  for  multiplication  of  fractions?  Solve  and  explain  the 
following:  Multiply  f  by  f. 

What  is  Case  I  in  division  of  fractions?  What  principle  underlies 
the  process?  Show  by  an  example  that  the  principle  is  true.  What 
is  Case  II?  Give  the  rule  for  dividing  an  integer  by  a  fraction. 
What  is  Case  III?  Solve  the  following:  What  is  the  value  of  |  -4-f  ? 
Give  an  analysis  and  explanation  of  the  process.  Give  the  general 
rule  for  division  of  fractions.  Describe  what  are  included  among 
fractional  forms,  How  are  they  simplified?  What  is  Case  I  in  frac- 
tional relation  of  numbers?  What  is  the  principle  upon  which  rela- 
tion of  numbers  is  based?  What  is  Case  II?  Illustrate  each  case  by 
an  example. 

What  is  a  decimal  fraction?  From  what  is  the  word  decimal 
derived?  How  are  decimal  fractions  expressed?  How  are  decimals 
distinguished  from  integers?  State  the  principles  of  decimal  frac- 
tions. Show  each  to  be  true.  What  is  the  decimal  point?  What 
other  name  has  it?  What  is  a  pure  decimal?  What  is  a  mixed  deci- 
mal? What  is  a  complex  decimal?  Name  the  orders  of  decimals  as 
far  as  ten-millionths.  How  does  the  place  occupied  by  any  order  of 
decimals  compare  with  that  occupied  by  integers  of  the  correspond- 
ing name? 

How  are  decimals  reduced  to  a  common  denominator?  Explain 
the  process.  How  are  common  fractions  reduced  to  decimals? 
Analyze  the  process.  If  a  common  fraction  can  not  be  exactly  re- 
duced to  a  decimal,  what  is  done?  How  do  addition  and  subtraction 
of  decimals  compare  with  the  same  processes  in  integers? 


366  TEST   QUESTIONS. 

What  is  the  principle  upon  which  multiplication  of  decimals  is 
based?  Show  that  it  is  true.  How  may  a  decimal  be  multiplied  by 
1  with  any  number  of  ciphers  annexed?  What  is  the  principle  upon 
which  the  process  of  division  of  decimals  is  based?  How  may  a  deci- 
mal be  divided  by  1  with  any  number  of  ciphers  annexed? 

How  may  we  multiply  by  a  number  that  is  a  little  less  than  a  unit 
of  the  next  higher  order?  How  may  we  multiply  when  one  part  of 
the  multiplier  is  a  factor  of  another  part?  How  may  we  multiply  by 
a  number  that  is  a  part  of  some  higher  unit?  What  is  an  aliquot  part 
of  a  number?  What  are  the  common  aliquot  parts  of  10?  What  of 
100?  How  is  the  cost  found  when  the  quantity  and  price  per  100  or 
1000  are  given? 

What  is  a  debt?  Define  what  is  meant  by  a  credit;  a  debtor;  a 
creditor;  an  account;  the  balance  of  an  account;  a  bill;  the  footing 
of  a  bill.  State  some  of  the  more  common  abbreviations  used  in 
business  correspondence. 

Tell  what  a  concrete  number  is;  an  abstract  number;  a  denominate 
number;  a  simple  denominate  number;  a  compound  denominate  num- 
ber; a  standard  unit;  a  scale.  Illustrate  each  of  the  preceding  by  an 
appropriate  example.  How  many  kinds  of  numerical  scales  are  there? 

What  is  money?  Of  how  many  kinds  is  it?  What  is  coin,  or 
specie?  What  is  paper  money?  Give  the  table  and  denominations 
of  the  currency  of  the  United  States.  What  are  the  ordinary  coins? 
What  are  the  denominations  and  coins  of  Canada?  Give  the  table 
of  English  money  and  the  coins  in  common  use.  What  are  the  cur- 
rency and  coins  of  France? 

What  is  meant  by  reduction  of  denominate  numbers?  What  is 
reduction  descending?  Give  the  rule.  What  is  reduction  ascending? 
Give  the  rule. 

Define  and  illustrate  what  is  meant  by  space,  a  line,  a  surface,  a 
solid.  For  what  are  linear  measures  used?  Repeat  the  table  of 
Linear  Measure,  and  of  Surveyor's  Linear  Measure.  What  is  an 
angle?  A  square?  A  square  inch?  A  rectangle?  What  is  the  area 
of  a  surf  ace  ?  How  is  the  area  of  a  rectangular  surface  computed? 
Repeat  the  table  of  Square  Measure,  and  of  Surveyors'  Square  Meas- 
ure. What  is  a  solid?  A  cube?  A  cubic  inch?  A  cubic  foot?  The 
volume  or  solid  contents?  How  is  the  volume  of  a  rectangular  solid 
computed?  Repeat  the  tables  of  Cubic  Measure,  and  Wood  and  Stone 
Measure. 


TEST   QUESTIONS.  367 

What  are  the  measures  of  capacity?  Kecite  the  table  of  Liquid 
Measure.  In  estimating  the  capacity  of  cisterns,  etc.,  how  many  gal- 
lons are  considered  a  barrel?  How  many  a  hogshead?  How  many 
cubic  inches  are  there  in  a  gallon?  Repeat  the  table  of  Apothecaries' 
Fluid  Measure.  For  what  is  Dry  Measure  used?  Repeat  the  table. 
How  many  cubic  inches  are  there  in  a  bushel? 

What  is  weight?  For  what  is  Avoirdupois  Weight  used?  Repeat 
the  table.  How  many  pounds  are  there  in  the  long  ton?  How  many 
grains  are  there  in  an  avoirdupois  pound?  For  what  is  Troy  Weight 
used?  Repeat  the  table.  How  many  grains  are  there  in  a  Troy 
pound?  For  what  is  Apothecaries'  Weight  used?  Repeat  the  table. 
How  many  grains  are  there  in  a  pound  Apothecaries'  Weight? 

Repeat  the  table  of  Measures  of  Time.  Explain  how  often  leap 
year  occurs.  What  is  a  circle?  What  is  the  circumference  of  a  circle? 
An  arc  of  a  circle?  A  degree  of  the  circumference?  What  is  the 
measure  of  an  angle?  Repeat  the  table  of  Circular  Measure.  What 
a  quadrant?  A  sextant?  Give  the  Stationers'  Table  and  the  table 
of  Counting.  Give  the  cases  in  Reduction  of  Denominate  Fractions. 
Solve  an  example  illustrative  of  each  case  and  explain  the  process. 

How  do  the  fundamental  processes  in  Compound  Denominate  Num- 
bers compare  with  the  same  processes  in  Simple  Numbers? 

How  does  the  number  of  degrees  apparently  passed  over  by  the  sun 
compare  with  the  number  of  hours  occupied  in  passing  that  distance? 
The  number  of  minutes  of  space  with  the  number  of  minutes  of  time? 
The  seconds  of  space  with  the  seconds  of  time?  Repeat  the  table 
showing  the  relation  between  longitude  and  time.  What  is  a  merid- 
ian? What  is  longitude?  Give  the  rule  for  finding  the  difference  in 
time  when  the  difference  in  longitude  of  two  places  is  given.  Give  the 
rule  for  finding  the  difference  in  longitude  of  two  places  when  their 
difference  in  time  is  given. 

What  is  the  unit  of  length  in  the  Metric  System  of  measures?  To 
what  is  it  nearly  equal?  What  is  the  metric  unit  of  area?  What 
the  unit  of  solidity?  What  the  unit  of  capacity?  What  the  unit  of 
weight? 

Define  per  cent.  What  is  the  commercial  sign  of  per  cent.?  Of 
what  does  Percentage  treat?  How  may  per  cent,  be  expressed? 
What  elements  are  involved  in  problems  in  Percentage?  What  is 
meant  by  the  base?  The  rate?  The  percentage?  The  amount?  The 
difference?  What  are  the  five  fundamental  problems  or  cases  in 


368  TEST   QUESTIONS. 

Percentage?  Solve  an  example  illustrating  each,  and  give  a  rule  for 
each  case. 

What  is  interest?  Define  the  terms  principal;  amount;  rate  of 
interest ;  legal  interest ;  usury  ;  a  note  or  promissory  note.  Give  three 
methods  for  computing  interest.  What  is  compound  interest?  Give 
the  rule  for  computing  compound  interest.  How  is  the  compound 
interest  table  formed?  What  is  meant  by  annual  interest?  Give  the 
rule  for  computing  annual  interest.  In  what  respect  does  compound 
interest  differ  from  annual  interest? 

What  are  partial  payments?  What  is  an  indorsement?  What  is 
the  Mercantile  Kule  for  computing  the  amount  due  when  partial  pay- 
ments have  been  made?  When  is  the  Mercantile  Kule  used?  What 
is  the  principle  upon  which  the  United  States  Rule  is  based?  Give 
the  United  States  Kule.  When  the  principal,  rate,  and  interest  are 
given,  how  is  the  time  found?  When  the  principal,  time,  and  interest 
are  given,  how  is  the  rate  found?  When  the  rate,  time,  and  interest 
are  given,  how  is  the  principal  found? 

What  is  a  promissory  note?  Who  is  the  maker  or  drawer?  Who 
is  the  payee?  Who  is  the  holder?  Who  is  the  indorser?  In  how 
many  ways  may  he  indorse?  What  is  the  face  of  a  note?  When  is  a 
note  negotiable?  When  is  a  note  not  negotiable?  What  are  days  of 
grace?  Write  a  negotiable  note  and  transfer  it  by  indorsement. 

What  is  discount?  What  is  commercial  discount?  What  is  net 
price?  What  is  the  cash  value  of  a  bill?  In  cases  where  there  is  a 
discount  of  some  per  cent.,  as  20 ^  off  and  5^,  off  for  cash,  upon  what 
sum  is  the  5^  computed?  What  is  true  discount?  Define  present 
worth.  Give  the  rule  for  solving  problems  in  true  discount.  What  is 
a  bank?  A  check?  Bank  discount?  The  proceeds  or  avails  of  a 
note?  The  maturity  of  a  note?  The  term  of  discount?  How  is  the 
bank  discount  computed?  Is  it  right  or  wrong  in  principle?  How 
can  we  find  how  large  to  make  a  note  that  we  may  have  a  certain  sum 
left  after  paying  the  discount  at  a  bank? 

What  elements  in  Profit  an<J  Loss  correspond  to  the  base,  rate,  per- 
centage, amount,  and  difference?  What  is  the  principle  upon  which 
computations  in  Profit  and  Loss  are  based? 

Define  the  terms  commission  merchant  or  agent;  commission;  a  con- 
signment; consignor;  consignee;  the  net  proceeds.  What  elements  in 
commission  correspond  to  base,  rate,  percentage,  amount,  and  differ- 
ence? Upon  what  principle  is  commission  based? 


TEST   QUESTIONS.  369 

What  is  a  tax?  What  is  real  estate?  What  is  personal  property? 
Wliat  is  a  property  tax?  What  is  a  personal  tax?  Who  is  an 
assessor?  What  is  an  assessment  roll?  Explain  how  taxes  are 
levied  and  the  individual  taxes  computed.  Explain  the  formation  of 
the  assessor's  table.  What  are  duties  or  customs?  What  is  meant  by 
specific  duty?  Ad  valorem  duty?  Tare?  Leakage  and  breakage? 
Custom-houses? 

What  is  meant  by  the  terms,  a  company ;  a  corporate  company  or 
corporation;  a  charter;  capital  stock;  a  share  of  stock;  a  certificate  of 
stock;  par  value;  above  par;  below  par;  market  value?  What  is  an 
installment?  What  is  an  assessment?  What  is  a  dividend?  De- 
scribe a  bond  and  coupons.  How  are  Government  securities  desig- 
nated? Name  the  various  classes  of  Government  securities,  and  state 
the  rate  of  interest  they  bear.  In  what  are  all  Government  bonds 
payable?  In  what  is  the  interest  of  all  Government  bonds  payable? 
What  are  stocks?  Who  is  a  stock-broker?  What  is  brokerage? 
What  elements  in  the  subject  of  stocks  correspond  to  base,  percentage, 
amount,  and  difference?  What  is  meant  by  the  expressions,  stock  is 
selling  at  83 J,  112,  etc.?  If  the  market  value  and  rate  of  premium  or 
discount  are  given,  how  can  the  par  value  be  found? 

What  is  insurance?  Of  how  many  kinds  is  it?  What  is  property 
insurance?  What  is  a  policy?  What  is  the  premium?  Of  how 
many  kinds  are  insurance  companies  with  regard  to  the  parties  who 
participate  in  the  profits?  What  is  a  mutual  insurance  company? 
What  is  a  stock  company?  What  is  a  mixed  company?  What  are 
the  elements  involved  in  insurance  that  correspond  to  base,  rate,  and 
percentage?  What  is  personal  insurance?  What  is  a  life  policy? 
What  an  endowment  policy?  What  an  accident  or  health  policy? 

What  is  exchange  ?  Explain  the  method.  What  is  a  draft  or  bill 
of  exchange  ?  Write  a  draft.  How  many  parties  are  there  connected 
with  a  draft  primarily?  Who  is  the  drawer?  The  drawee?  The 
payee?  What  is  a  sight  draft?  A  time  draft?  What  is  meant  by 
accepting  a  draft  ?  Of  how  many  kinds  is  exchange  ?  What  is  do- 
mestic exchange?  State  and  solve  examples  illustrating  the  rules. 
What  is  foreign  exchange  ?  What  is  a  set  of  exchange  ?  Upon  what 
cities  in  Europe  are  drafts  more  commonly  drawn?  What  is  the 
value  in  United  States  gold  coin  of  a  sovereign  ?  What  is  the  value 
of  a  franc  ?  Solve_  an  example  illustrating  the  principles  of  foreign 
exchange. 

24 


370  TEST   QUESTIONS. 

What  is  meant  by  averaging  payments?  The  average  time?  The 
term  of  credit?  The  average  term  of  credit?  Solve  an  example  illus- 
trating the  process  of  solution  when  the  terms  of  credit  begin  at  the 
same  time.  Solve  an  example  when  the  terms  of  credit  begin  at  dif- 
ferent dates.  Explain  the  process.  Explain  the  process  of  averaging 
accounts,  and  give  the  rule. 

What  is  partnership?  Who  are  partners?  What  is  the  capital  of 
a  firm  or  company?  Give  the  principle  underlying  partnership. 
What  is  Case  I  in  partnership?  What  is  Case  II?  Solve  and  analyze 
an  example  in  Case  II.  Give  the  rule  for  partnership  settlements. 

What  is  ratio?  Of  how  many  kinds  is  it?  What  are  the  terms  of 
a  ratio?  The  antecedent?  The  consequent?  What  is  the  sign  of 
ratio?  From  what  may  it  be  regarded  as  derived?  What  is  a  coup- 
let? What  are  the  principles  of  ratio?  Illustrate  each  principle. 

What  is  proportion?  What  is  the  sign  of  proportion?  From  what 
may  it  be  regarded  as  derived?  What  are  the  antecedents  of  a  pro- 
portion? The  consequents?  The  extremes?  The  means?  Give  the 
principles  of  proportion  regarding  the  relation  of  extremes  and  means. 
What  is  meant  by  a  simple  ratio?  A  simple  proportion?  A  direct 
proportion?  An  inverse  proportion?  Give  the  rule  for  solving  ex- 
amples in  simple  proportion.  What  is  a  compound  ratio?  A  com- 
pound proportion?  The  principle  underlying  compound  proportion? 
Solve  an  example  in  compound  proportion,  by  successive  simple  pro- 
portions and  by  cause  and  effect.  Give  the  rule  for  compound  pro- 
portion. 

What  is  a  power?  How  are  powers  named?  What  is  an  exponent? 
What  is  involution?  How  is  the  power  of  a  number  obtained?  How 
is  the  square  of  a  number  expressed  in  terms  of  its  tens  and  units? 
How  in  terms  of  any  two  parts?  How  is  the  cube  of  a  number  ex- 
pressed in  terms  of  its  tens  and  units?  How  in  terms  of  any  two 
parts? 

What  is  a  root?  How  are  roots  named?  What  is  evolution? 
What  is  the  radical  or  root  sign?  Define  a  perfect  power;  an  im- 
perfect power.  Give  the  rule  for  finding  the  root  of  a  number  by 
factoring.  Give  the  principle  relating  to  the  number  of  figures  re- 
quired to  express  the  square  of  a  number;  the  cube  of  a  number. 
Give  the  principle  relating  to  the  number  of  figures  in  the  square 
root  of  a  number;  the  cube  root  of  a  number.  Solve  and  explain  an 
example  in  square  root.  Repeat  the  rule.  What  is  done  when  the 


TEST   QUESTIONS.  371 

number  is  not  a  perfect  square?  How  are  decimals  pointed  off? 
How  is  the  square  root  of  a  common  fraction  found?  How  is  the  side 
of  a  square  found  when  its  area  is  given? 

What  relation  do  the  squares  described  upon  the  sides  of  a  right- 
angled  triangle  sustain  to  each  other?  How  is  the  hypotenuse  of  a 
right-angled  triangle  found  when  the  other  sides  are  given?  How  is 
either  side  found  when  the  hypotenuse  and  the  other  side  are  given? 
What  are  similar  figures?  What  is  the  relation  between  similar 
surfaces  ? 

Solve  an  example  in  cube  root.  Deduce  from  your  solution  a  rule. 
What  is  done  when  there  is  a  remainder  after  subtracting  the  last 
product?  How  are  decimals  pointed  off?  How  is  the  cube  root  of 
a  common  fraction  found?  Give  the  principles  relating  to  similar 
solids. 

Define  a  series;  an  ascending  series;  a  descending  series;  an  arith- 
metical progression;  a  geometrical  progression.  How  is  the  sum  of 
an  aritmetical  series  found?  Illustrate  by  an  example.  How  is  the 
sum  of  a  geometrical  series  found?  Illustrate  by  an  example. 

Define  mensuration;  a  line;  a  straight  line;  a  curved  line;  paral- 
lel lines;  a  plane  surface;  a  curved  surface;  an  angle;  a  right  angle; 
an  acute  angle;  an  obtuse  angle;  a  vertex  of  an  angle;  a  triangle; 
a  quadrilateral;  a  parallelogram;  a  rectangle;  a  polygon;  a  circle; 
the  circumference  of  a  circle;  a  radius  of  a  circle;  a  diameter  of  a 
circle;  the  base  of  a  figure;  the  altitude  of  a  figure;  a  diagonal  of  a 
figure;  the  perimeter  of  a  figure;  the  area  of  a  surface. 

How  is  the  circumference  of  a  circle  obtained  from  its  diameter? 
The  diameter  from  the  circumference?  How  is  the  area  of  a  paral- 
lelogram computed?  How  is  the  area  of  a  triangle  computed?  Give 
the  rule  for  computing  the  area  of  a  trapezium;  a  trapezoid;  a  regu- 
lar polygon;  a  circle. 

What  is  a  solid?  A  prism?  A  parallelopipedon?  A  cylinder? 
A  pyramid?  A  cone?  A  frustum?  A  sphere?  A  diameter?  A 
radius  of  a  sphere?  The  circumference  of  a  sphere?  The  altitude 
of  a  solid?  The  convex  surface  of  a  solid? 

Give  the  rule  for  finding  the  convex  surface  of  a  prism  or  cylinder; 
a  pyramid  or  cone;  a  frustum  of  a  pyramid  or  cone;  a  sphere.  Give 
the  rule  for  finding  the  volume  of  a  prism  or  cylinder;  a  pyramid  or 
cone;  a  frustum  of  a  pyramid  or  cone;  a  sphere. 


-^saa^^                  nnm^ 

Page  28. 

Page  31. 

18.  182. 

2.  947. 

34.  42151. 

19.  189. 

3.  1298. 

35.  118422. 

20.  ,367. 

4.  $42.70. 

36.  12149865. 

21.  186. 

5.  $28.48. 
6.  10845. 
7.  20839. 

37.  $5165000. 
38.  5871  period. 
20842475  circ. 

22.  93. 
23.  865. 
24.  3886. 

8.  6283. 

25.  5828. 

9.  3530. 

Page  32. 

26.  2131. 

10!  21974. 
11.  20002. 
12.  $103.485. 
13.  $1437.845. 
14.  933324. 

39.  $1595.85,  Troy; 
$4807.37,  N.Y.; 
$6403.22,  both. 
40.  45764. 
41.  47388. 

27.  8132. 
28.  10570. 
29.  20068. 
30.  28879. 
31.  80091. 

15.  134739. 

42.  40590. 

32.  8486888. 

16.  349950638. 

43.  540356. 

33.  256  mi. 

17.  $132.94. 

44.  8882. 

34.  $508. 

18.  1600. 

45.  11031. 
46.  109546. 

Page  42. 

Page  29. 

47.  86672. 

35.  1163. 

19.  $14593. 
20.  21502. 

Page  41. 

36.  $131. 
37.  1955. 

21.  9265. 

2.  305. 

38.  609973105. 

22.  2775. 

3.  228. 

39.  159  acres. 

23.  8692. 

4.  292. 

40.  $202.12. 

24.  934. 

5.  272. 

41.  1492. 

25.  25917. 

6.  879. 

42.  $1697. 

7.  61. 

43.  $9036. 

Page  30. 

8.  919. 

44.  8687. 

26.  997. 

9.  288. 

45.  3502. 

27.  645621. 
28.  528408. 

10.  1299. 
11.  150. 

Page  48. 

29.  $8484. 

12.  $3.07. 

2.  942. 

30.  7005  lib.; 

13.  $11.26. 

3.  2272. 

1999881  vol. 

14.  $16.07. 

4.  3948. 

31.  402399. 

15.  $23.96. 

5.  1725. 

32.  377805. 

16.  $23.67. 

6.  $4095. 

33.  642502. 

17.  289. 

7.  $2307. 

(372) 

ANSWERS. 


373 


8.  1570. 

26.  47913214. 

4570500. 

9.  2620. 

27.  66319209. 

7.  1215000; 

10.  5348. 

28.  92317008. 

15552000; 

11.  2490. 

29.  76526998. 

20412000; 

12.  4404. 

30.  82613817. 

31590000. 

13.  $432. 

31.  143555004. 

8.  655200; 

14.  $336. 

32.  291006415. 

11856000; 

15.  $192. 

33.  281072720. 

8424000  ; 

16.  $399. 

34.  1151707808. 

14352000. 

17.  $138. 

35.  570873555. 

18.  $16555.'      x 

36.  1381600350. 

Page  55. 

19.  $33.75. 

37.  1251756036. 

9.  2640000. 

20.  $250.16. 

38.  1253542212. 

10.  48000. 

Page  49. 

Page  53. 

11.  $832. 
12.  $18256. 

21.  3825. 

39.  $1358.53. 

22.  1568. 

40.  $5211.45. 

1.  $32739.84. 

23.  $16.02. 

41.  $13309.92. 

2.  $13730.50. 

24.  $43.75. 

42.  $99253.80. 

3.  1229. 

25.  36960. 

43.  $152323.35. 

4.  339904. 

26.  $300. 

44.  $2332.99. 

5.  $4299. 

27.  $127.75. 

45.  $69520.33. 

6.  $168. 

28.  $668. 

46.  $69577.43. 

7.  10293. 

47.  $213469.56. 

8.  4776. 

Page  52. 

48.  $262816.86. 

9.  $35.90. 

2.  13650. 

49.  58650. 

3.  11826. 

50.  11024. 

Page  56. 

4.  17664. 

51.  260520. 

10.  89232. 

5.  $64.63. 

52.  5022. 

11.  $247.52. 

6.  $114.48. 

53.  $48535. 

12.  $169.25. 

7.  5472. 

54.  $35105. 

13.  10248. 

8.  8947. 

55.  $5108066. 

14.  108  mi. 

9.  12325. 

15.  $635.14. 

10.  10098. 

Page  54. 

16.  $17481.30. 

11.  101500. 

3.  3750; 

17.  $2066.96. 

12.  74803. 

37500; 

18.  $1563.25. 

13.  231903. 
14.  3485092. 

15000; 
112500. 

Page  64. 

15.  3942146. 

4.  25350; 

11.  1218. 

16.  6568848. 

59150; 

12.  1366. 

17.  1344455. 

507000; 

13.  496. 

18.  1332675. 

760500. 

14.  597. 

19.  585575. 

5.  88000; 

15.  1545. 

20.  1570688. 

123200; 

16.  984. 

21.  3334932. 

70400; 

17.  406. 

22.  2546650. 

176000. 

18.  876. 

23.  23748222. 

6.  277000; 

19.  554. 

24.  25721944. 

2770000  ; 

20.  1353. 

25.  62748374. 

3047000  ; 

21.  548. 

374 


ANSWERS. 


22.  1234. 

23.  913f. 

24.  1488f. 

25.  908 1. 

26.  2296|. 

27.  1205k 

28.  485|. 

Page  67. 

2.  124. 

3.  313. 

4.  216. 

5.  306. 

6.  406. 

7.  432. 

8.  416. 

9.  46. 

10.  312. 

11.  442. 

12.  41. 

13.  22. 

14.  77. 

15.  103. 

16.  213. 

17.  119. 

18.  131. 

19.  114. 

20.  141. 

21.  67. 

22.  505. 

23.  315. 

24.  406. 

25.  519. 

26.  525. 

27.  541. 

28.  606. 

29.  723. 

30.  544. 

31.  752. 

32.  664. 
38.  777. 

34.  1802. 

35.  1945. 

36.  4372. 

37.  6203. 

38.  9216fff. 

39.  5679JJ 
40. 

41. 


42.  81987AV 

30.  $3837.50. 

43.  771849f£f. 

31.  $92278  if|f. 

44.  474536T5o2iV 

32.  18  minutes. 

45.  252384  f  iff. 

33.  105  head. 

46.  12752fif4f. 

34.  10  grandchild'n. 

47.  854104|f  J|f. 

35.  3716|gg§  Ib. 

36.  21f  Jiff  acres. 

Page  68. 

48.  162. 

Page  79. 

49.  $258. 

2.  7,  5. 

50.  130  ini. 

3.  26. 

51.  18  doz. 

4.  2*,  3,  7. 

52.  I093fjf  J. 

5.  23,  3,  7. 

53.  5|||§. 

6.  24,  32. 

54.  140  da.  480  left. 

7.  32,  5,  7. 

55.  494iifgg§. 

8.  2,  32,  11. 

56.  $2029}. 

9.  25,  7. 

57.  $100863%. 

10.  2,  3,  131. 

58.  5280. 

11.  22,  79. 

59.  100. 

12.  22,  II2. 

60.  21-gtfffa. 

13.  28,  5. 

14.  24,  32,  7. 

Page  70. 

15.  22,  3,  5,  19. 

16.  25,  37. 

3.  ISy^. 

17.  24,  32,  13. 

&'.  127IM. 

18.  22,  3,  72,  13. 
19.  2,  3,  5,  7,  11. 

6.  39m. 

20.  22,  32,  89. 

7-  2T4o%V 

21.  32,  52,  7. 

"•  "4auog 

22.  7,  13,  43. 

9.  75T5T°o57. 

23.  22,  953. 

10.  50^.^ 

24.  72,  11,  13. 

12!  118;54964bu.rem. 
13.  640. 

25.  22,  5,  199. 
26.  23,  5,  11,  61. 
27.  22,  3,  11,  172. 

Page  73. 

28.  22,  34,  5,  7. 
29.  23,  3,  7,  11,  13. 

19.  2546-f. 

30.  22,  53,  37. 

20.  $42if-4-f. 

31.  29,  35. 

21.  SlyV^j  wk. 

22.  $60. 

Page  SO. 

23.  1920  bn. 

6.  13600; 

24.  $2  per  bbl. 

15300; 

25.  45  yd. 

20400; 

26.  500  mi. 

30600. 

27.  20  da. 

7.  102144; 

28.  80  bu. 

49248  ; 

29    60  acres  ; 

82080; 

$400  loss. 

196992. 

ANSWERS. 


375 


8.  $2240. 

Page  86. 

4.  630. 

9.  $112.35. 

2.  4. 

5.  144. 

10.  $12096. 

3.  9. 

6.  300. 

11.  $225. 

4.  12. 

7.  540. 

12.  $2208. 

5.  6. 

8.  770. 

13.  $1320. 

6.  12. 

9.  720. 

14.  $1620. 

7.  9. 

10.  960. 

15.  $131.60. 

8.  16. 

11.  2835. 

9.  9. 

12.  6650. 

Page  81. 

10.  7. 

13.  288. 

9.  71. 

11.  8. 

14.  2240. 

10.  315. 

12.  12. 

15.  700. 

11.  205. 

13.  5. 

16.  41580. 

12.  34. 

14.  6. 

17.  401115. 

13.  23. 

15.  7. 

14.  49. 

16.  4. 

Page  91. 

15.  119J1. 

17.  11. 

19.  360. 

16.  44||. 

18.  24. 

20.  1620. 

17.  58|j. 

19.  42. 

21.  240. 

18.  121/T. 

22.  1440. 

1Q      Q1    9 

Page  87. 

23.  900. 

20*.  304A, 

2.  13. 

24.  85800. 

&  4 

3.  11. 

25.  48. 

Page  82. 

4.  12. 

26.  480. 

21.  224  canisters  ; 
32  packages. 
22.  152  packages  ; 
38  quires. 

5.  17. 
6.  27. 
7.  23. 
8.  42. 
9.  52. 

27.  1200. 
28.  20160. 
29.  240. 
30.  24  inches. 
31.  60  yards. 

Page  83. 

10.  27. 
11.  21. 

32.  120. 
33.  210  bu. 

3.  7. 

12.  34. 

34.  360. 

4.  2. 

13.  4. 

5.  3. 

14.  32. 

Page  92. 

6.  2|. 

15.  17. 

35.  120  days. 

7.  4. 

16.  33. 

36.  436T\. 

8.  3f. 

17.  126. 

37.  12  feet. 

9.  4. 

18.  42. 

38.  4. 

10.  4i. 

19.  37. 

39.  5275. 

11.  10. 

20.  14  in. 

40    1728. 

12.  186f. 

/1.  $336. 

Page  si. 

Page  88. 

21.  5  Ibs. 

42.  2025. 
43.  8  rods. 

13.  7. 

22.  67  fields; 

44.  41  f. 

14.  16425. 

6  acres  each. 

45.  7. 

15.  107800. 

46.  378. 

16.  $43|f. 

Page  9O. 

17.  7iJ  bu. 

2.  448. 

Page  98. 

18.  $3.12. 

3.  144. 

2.  !§. 

376 


ANSWERS. 


Page  10O. 


6. 

7. 


Page  1O1. 


6    4-6- 

u'       9 

7.  !» 

8.  -W-;  - 


9.  W-; 

10.  ija. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

25. 


Page  103. 


2.  $5. 

O  -I  O_2 

4!  io|.T< 

5.  16f 


7. 


10. 

11.  27jf 

13!  6i||.3' 

14.  18J. 

15.  14^%. 

16.  181?. 

17.  129||f. 

18.  843225?- 

19.  58TVV 

20.  97f|J. 

21.  16TVW. 

22.  6|H*- 


Page  104. 


10. 
11. 

12-  fs-6-;  tt5i  ft*- 
13.  MV;i%;iW 

14. 


Page  108. 


Page  109. 


ANSWERS. 


377 


20.  52111 

323T«T; 

11.  TJ¥. 

21.  881. 

370. 

12.  ?V 

22.  $3.      - 
23.  222i  sold; 
189|  left. 

Page  114. 

13.  ^. 
Page  117. 

24.  $19|. 

3.  Jf. 

15.  2||. 

4     ijL 

16.  6T\. 

Page  HO. 

O.    jj. 

17.  4f|. 

3   l^' 

6.     A' 

18.  2Ji 

7.  i. 

19.  3JJ. 

i  if 

5.  -fj. 

8.  A. 
9.  if. 

20.  3fi 
21.  i 

6    21TV. 

10.  A* 

22.  rf  o- 

7l  6. 

11.    Tfy. 

23.  $4ff. 

8.  f. 

12-  2"linj' 

24.  $|. 

9.  If. 

13.  T|o- 

on      d^i  -i    5 
^O.    «]pll^2". 

10.  21 

14.  A 

26.  $1072^77- 

11.  31. 

15.  i|f. 

12.  4if. 

16.  H- 

Page  118. 

13.  51 

17.  /7. 

2.  42. 

14.  5A- 

18.  rffr. 

3.  72. 

15.  8|f. 

19.  |. 

4.  llj. 

16.  14T2T. 

20.  A- 

5.  56. 

21.  ff. 

6.  80. 

Page  111. 

22.  i 

7.  861 

17.  $5f. 

24.  3A- 

8.  60. 

18.  $3. 

25.  27. 

9.  130. 

20.  86f. 

26.  311 

10.  120. 

21.  $841 

27.  147 

11.  40f. 

22.  154f. 

28.  $|. 

12.  191 

29.  90|. 

13.  931. 

Page  112. 

•i 

30.  f  . 

14.  49. 

2.  31. 

31.  534  1. 

15.  68. 

3.  2|. 

32.  $81. 

16.  69. 

4.  4|. 

17.  123. 

5.  61. 

Page  115. 

18.  60. 

6.  12. 

33.  1AV- 

19.  111. 

7.  99. 

34.  32f|f 

20.  305. 

8.  15. 

21.  145. 

9.  24if. 

Page  116. 

22.  128. 

10.  $76A- 
11.  $87. 

2-  A- 
3.  i 

23.  63T93. 
24.  482-V 

12.  $549A- 
13.  $421f. 
14.  $4710. 

4.  A- 
5.  7J. 
6.  /f. 

25.  21T3f. 
Page  119. 

15.  88. 

T  3' 

27.  4?, 

17.  198; 

8      /-r. 

28.  3A- 

237f; 

9.  A- 

29.  3||. 

272; 

10.  7y 

30.  5|f. 

378 


ANSWERS. 


31.  4ff. 
32.  6T7T- 

12.  if. 
13.  11TV 

14.  $894. 
15.  16  bu. 

33.  20  yd. 

14.  ?\. 

16.  ij. 

34.  21  J  bu. 
35.  20. 

15.  J. 
16.  6f. 

Page  127. 

36.  $9}|. 

17.  A. 

17.  222. 

38!  f  If'lb. 

Page  124. 

18.  $6  Iff. 
19.  f 

1  Q       3 

TiO"* 

20.  -5-  A's  or  $4363 

Page  121. 

20.  A. 

'  4f'B's,'or$8726 

7.  f 

21.  TV 

21.  $105. 

8.  if. 

22.  A. 

22.  $3ff. 

9.  7. 
10.  if. 

23.  ?V 
24.  Tf^. 

23    1219 
24!  $11354f. 

11.  f. 

25.  ^V 

25.  $159861. 

12.  1-fa. 

26.  ^V 

26.  $24TV7,  first; 

13.  li£. 

27.    A. 

$28^^  second. 

15.  1T\. 

29.  f. 

27.  $2600,  one's; 

16.  fo. 

30.  f  f. 

$3900,  other's. 

17.  2J. 

31.  f£. 

28.  $5797i. 

18   -7- 

32.  ff. 

J-v-'*     66* 

19.  2f|. 

33.  H- 

Page  128. 

20.  2T\. 

34.  f|. 

29.  TV,  A; 

21.  H- 

35.  ft. 

22.  7TV 

36.  |?. 

?%,  both  ; 

23.  3|. 

37.  |J. 

4f  da. 

24.  26^. 
25.  93|f. 

Page  125. 

30.  3TV  da. 
31.  30  da. 

13.  if-;   V- 

32.  40,  shorter; 

Page  122. 

14.  1  ;  if. 

84,  longer. 

26.  46. 

15.  TV;  1. 

33.  $43. 

27,  7|. 

16.  f  ;  J. 

34.  26T87  da. 

28.  8/7. 

35.  45ft.;  30ft. 

29.  If. 
30.  13J|. 
31.  T\. 

2*.  $7L30. 
3.  2J  acres. 

36.  60  and  80. 

37.  $8269i 
38.  $2700,"A; 

32.  1689  f  bu. 

Page  126. 

$3000,  B. 
39.  $48. 

Page  123. 

4.  64^  cents. 

40.  28  da.,  A; 

2.  A. 

5.  6|. 

21  da.,  B. 

O.     j^j. 
4.    yV 

7!  37|.8£ 

Page  136. 

O.    J.yg'. 

8.  $925^  gain. 

9       27 
*•    1TOO"* 

6.  TV 

9.  1632-V 

7.  TV. 

10.  206  6  \\. 

4      —  — 

8.  7f. 

11.  153i. 

5i.  yC' 

8 

9.  1^. 

12    9— 

6    4  l- 

10.  fi. 

13.'  $2c850,    son; 

7.  M&. 

11.  2;^. 

$2533  J,  daughter.       8.  ^^. 

ANSWERS. 


379 


Page  137. 

2.  .25. 

3.  .6. 

4.  .625. 

5.  .375. 

6.  .8. 

7.  .75. 

8.  .0625. 

9.  .15. 

10.  .85. 

11.  .52. 

12.  .35. 

13.  .3833  +  . 

14.  .555  +  . 

15.  .3846  +  . 

16.  .233  +  . 

17.  .1406  +  . 

18.  2.625. 

19.  .024. 

20.  .1875. 

21.  .03907  +  . 

22.  .42857  +  . 

23.  .4545  +  . 

24.  .3157  +  . 

25.  .23809  +  . 

26.  15.625. 

27.  24.6. 


28.  .825. 

4.  .20056. 

29.  3.425. 

5.  153.12. 

30.  .23625. 

6.  2.2272. 

31.  .625. 

7.  25.752. 

32.  .875. 

8.  74.375. 

33.  .4375. 

9.  .020265. 

34.  4.2155  +  . 

10.  1822.5. 

35.  37.54. 

11.  5.8776. 

36.  20.06. 

12.  15.1296. 

37.  .0001625. 

13.  34.4576. 

Page  13  8. 

14.  $23.375. 
15.  $167.485. 

2.  8.497. 

16.  122.994. 

3.  8.7907. 

17.  28.0685. 

4.  8.914. 

18.  $10.12. 

5.  72.379. 

19.  2680.804. 

6.  .40035. 

20.  1137.19424. 

7.  117.766. 

21.  2.837025. 

Page  139. 

22.  145.81944. 
23.  4.0596288. 

8.  $108.455. 

24.  272.80767. 

9.  $68.19. 

25.  1.0725. 

10.  $5394.267. 

27.  38464; 

11.  30.975. 

3846.4; 

12.  $32.1875. 

384640. 

13.  .850955. 

28.  184.65; 

14.  $77.7155. 

1846.5; 

15.  $1102.345. 

18465. 

16.  $48.365. 

29.  $108.675. 

Page  140. 

Page  143. 

2.  24.903. 

30.  $5212.84. 

3.  25.964. 

31.  391.05  rd. 

4.  $24.875. 

32.  $34631.37. 

5.  $74.875. 

33.  $60.6125. 

6.  20695. 

34.  $477.375. 

7.  2.0232. 

35.  $1706.16. 

8.  99.96154. 

36.  $71.445. 

9.  .0000756. 

10.  79999.92. 

Page  145. 

11.  $11.94. 

2.  .25. 

12.  $17.705. 

3.  .005. 

13.  $4.105 

4.  1.6895  +  . 

14.  $2519.98. 

5.  .00365. 

15.  $76468.06. 

6.  76.3. 

7.  30.2. 

Page  142. 

8.  2.13. 

2.  .2210. 

9.  .15. 

3.  2.025. 

10.  3650. 

380 


ANSWEKS. 


11.  27500. 

4.  6700. 

10.  277  J  Ib. 

12.  2643.6923  +  . 

5.  126400. 

11.  1568.96  +  lb. 

13.  .21. 

6.  2521000. 

12.  $292.25. 

14.  10020. 

7.  420200. 

13.  $437.50. 

15.  .84472  +  . 

8.  81000. 

14.  5200  Ib. 

17.  .4826. 

9.  282400. 

15.  2000000. 

18.  .382457. 

10.  546600. 

16.  .00000002. 

19.  .0138542. 

11.  $6.75. 

17.  25.818  +  da. 

20.  .04897. 

12.  $274.66|. 

18.  $174.725. 

21.  .00006045. 

13.  $162. 

19.  102.56  +  wk. 

22.  .384563. 

14.  $107.625. 

20.  $689.83|. 

23.  3.5575  +  T. 

15.  $246.25. 

21.  $14449.50. 

24.  180doz. 

16.  $438.625. 

22.  $4.728. 

25.  $3.04. 

23.  $100000. 

26.  23  hhd. 

Page  15O. 

24.  .2169  +  . 

27.  37  stoves. 

2.  $21.26^. 

25.  $.75TV 

3.  $40.89  +  . 

26.  .35. 

Page  146. 

4.  $9.14  +  . 

27.  241.95,    A; 

2.  381744. 

5.  $755.771. 

120.975,  B; 

3.  4822336. 

6.  $356.85  +  . 

282.275,  C. 

4.  48518433. 

7.  $22.668  +  . 

5.  8794128. 

8.  $41.06J. 

Page  161. 

6.  345684825. 

9.  $18.645. 

2.  606d. 

7.  47235605. 

10.  $72.875. 

3.  265s. 

8.  80952576. 

11.  $71.64. 

4.  4148  far. 

9.  5862014256. 

12.  $56.25. 

6.  90d. 

13.  $63.085. 

7.  210d. 

Page  147. 

14.  $264.158. 

8.  26f  far. 

10.  $373.45. 
11.  $1318.08. 
12.  $13483. 

Page  153. 

2.  $293.06.  . 
3.  $2935.00. 

9.  lO^d. 
10.  106s.; 
5088  far. 
11.  598  far. 

2.  240975. 

12.  1793d. 

3.  555489. 

Page  154. 

13.  843  far. 

4.  999498. 

4.  $2393.60. 

14.  7s.  6d. 

5.  1474812. 

5.  $26.20. 

15.  5436  far. 

6.  3511158. 

6.  $921.36. 

16.  8480d. 

7.  9398184. 

7.  $181.88. 

17.  43383  far. 

8.  14850395. 

18.  28724  far. 

9.  1318044. 

Page  155. 

10.  1583232. 

1.  3631  ib. 

Page  163. 

11.  1564191. 

2.  127.16975  Ib. 

2.  7s.  2|d. 

12.  2415987. 

3.  2075  f|  sq.  ft. 

3.  £22  16s. 

13.  10158948. 

4,  36.153"+cu.  ft. 

4.  £6  12s.  2d. 

14.  87855880. 

5.  2150420. 

5.  £4  Os.  6d. 

15.  107378577. 

6.  1000. 

7.f     3 
*         1  0  0  * 

7.  10000000. 

O.     TT'jS. 

Page  149. 

8.  261907.8. 

9.  £f  12s. 

3.  8600. 

9.  $16.36£. 

10.  £157  6s. 

ANSWERS. 


381 


11.  £14  19s.  8d. 

Page  17O. 

1.  24  ft. 

12.  80s  lid. 
13.  £242  l6s. 

16.  88  sq.rd.  26  sq.yd. 
8  sq.  ft. 

2.  13J  ft. 
3.  10|  ^ 

14.  £6  4s.  4d  . 

17.  f  ;  i;  f. 

4.  16J  ft. 

15.  £16  12s. 
16.  97s.  7Jd. 

18.  11  sq.  yd.  3  sq.  ft. 
13^  sq.  in. 

Page  175. 

17.  £50  11s.  9fd. 

19.  77  sq".  in. 

5.  180ft. 

18.  3572  far. 

20.  120  sq.  ft. 

6.  $37.312. 

19.  £50  12s.  5d. 

21.  42  sq.  yd. 

7.  $53.90. 

20.  14784  far. 

22.  25  sq.  it. 

8.  $27.561. 

21.  $72.9975. 

23.  48  sq.  yd.: 

9.  $21.09f. 

22.  £93  14s.  i-d. 
23.  £81  Is.  Id. 

M.     J         ) 

$55.20,  cost. 
24.  34yd. 

Page  176. 

24.  £7  14s.  IJd. 

25.  $79.80. 

4.  296  pt. 

25.  $121.66J. 

26.  165ft.  long; 

5.  $17.10; 

26.  14880  far. 
27.  £200. 

$35392.50,  cost. 
27.  80  rd. 

85i  gai. 
6.  120  gal.  Ipt.  2gi. 

28.  £250. 

608  gal.  2  qt.  1  pt. 

Page  171. 

7.  135  gi. 

Page  166. 

28,  $472.50. 

8.  4224  gi.   60  gal. 

13.  2  mi. 

29.  90  sq.  ft. 

2qt. 

14.  8903  yd. 

30.  $35.20. 

9.  192  pt.  8  bbl.  12 

15.  1452  in. 

31.  $48. 

gal.  1  qt.  3  gi. 

16.  443520  in.; 
570240  in. 
17.  19T9T  rd. 

32.  85  j  sq.  yd.,  sides; 
34  sq.yd.,  ceiling; 

$44.23,  cost. 

10.  1617. 
11.  16^. 
12.  284    bbl.    30.623 

18.  186  rd. 

33.  $21.39. 

gal. 

19.  12|  mi. 

20.  952204  in. 

Page  173. 

Page  177. 

21.  509716  in. 

1.  3456;  5184; 

4.  251  pt. 

22.  1  mi.  15  rd.  3  yd. 

25920;  55296. 

5.  525  pt. 

1  ft.  4  in. 

2.  54;  81;  351;  675. 

6.  526  bu,  1  pk.  5  qt. 

23.  2  mi.  162  rd.  4  yd. 
2ft. 

3,  640;  1024. 
4.  16  1  perch  ; 

Page  178. 

Page  167. 

$29.56,  cost. 

7.  218  bu.  6  qt.  1  pt. 
8.  1792bu.2pk.4qt. 

24.  41775360  ft. 

6.  207JI  yd. 

9.  2301  pt. 

25.  5  ft. 

7.  55410  cu.  in. 

10.  15052.8;  17203.2; 

26.  271  rd.  8  1. 

8.  369  cu.  ft. 

21504;  43008. 

27.  47322  in. 

9.  384|  cu.  ft. 

11.  6.4388  +  ; 

Page  169. 

10.  13824  blocks. 
11.  157i  cu.  ft. 

16.6336  +  ; 
22.6446+, 

10.  12111  sq.  in. 

12.  483840  cu.  in.; 

11.  12044056860  sq.  in. 

Page  174. 

225  bu. 

12.  6  A.   4  sq.  rd.  26 

12.  $10.311. 

13.  260.357  +  . 

sq.  yd.  2  sq.  ft. 
13.  14003316  sq.  in. 

13.  225  bu. 
14.  $348.44,  excav'g  ; 

Page  179. 

14.  4  sq.  rd.  21  sq.  yd. 

$418.18,  wall. 

4.  6205. 

1  sq.  ft.  89  sq.  in. 

15.  44517. 

5.  10216. 

382 


ANSWERS. 


6.  $10.875. 

Page  186. 

8.  ¥rg. 

7.  $2.52. 

2.  288;  492. 

95 
'     43"' 

8.  $42.24. 

3.  20736. 

10.  f^v 

Page  180. 

4.  $16.20. 
5.  70  yr. 

Page  19O. 

9.  100;  70;  98;'  25. 

6.  $9. 

12.  .17708  +  da. 

10.  $4.165. 

13.  .8125  bu. 

11.  $325. 

Page  187. 

14.  .2121  +  rd. 

12.  $9.885. 

8.  7  oz.  4  pwt. 

15.  £|f  J. 

13.  54.88  Ib. 
14.  $54.32TV 

9.  8cwt.  571b.  2f  oz. 
10.  26  rd.  3  yd.  2  ft. 

.  16.  .9239  +  £. 
17.  yVoV  cwt. 

15.  $12. 

11.  43  sq.  rd.   19  sq. 

18.  .1183  +  mi. 

16.  982f. 

yd.  2  sq.  ft.  36 

19.  .4597  +  wk. 

17.  54  bags. 

sq.  in. 

20.  .4312  +  reams. 

Page  181. 

12.  6  qt.  1  pt.  1J  gi. 
13.  7  hr.  12  min. 

21.  .4296  +  C. 
22.  .3717  -f  mi. 

3.  3498. 

14.  18  sq.  yd.  8  sq.  ft. 

23.  l^i  Ib. 

4.  7  oz.  4  pwt. 

22  J  sq.in. 

5.  $74.25. 
6.  8  spoons;  1  oz.  left. 

15.  9cu.ft.  777|cu.in. 
17.  |f  pt. 

1.  $4.805. 

2.  $2.34. 

7.  564  powders. 

18.  f  ft. 

3.  $19.68|. 

Page  183. 

19.  f£sc. 
20.  .384  pt. 

4.  $525.50. 
5.  21  Ib.  4  oz. 

8.  18912. 

c 

6.  $12.375. 

9.  23258. 

Page  188. 

7.  11JC.; 

10.  13  hr.  31  min.  35 

22.  11s.  6d. 

sec. 
11.  10  hr.  41  min.  37 

23.  2  oz.  6  pwt.  10.56 

8.  7.01822ft. 

gr. 

Page  191. 

sec. 
12.  2628000. 
13.  120  da. 
14.  197  da. 
15.  469  hr. 

24.  8qr.  17.28  sh. 
25.  1  pt.  1.904  gi. 
26.  2  ft.  2.73  in. 
27.  145  rd.  9  ft.  10.8 
in. 

9.  $.1772+  per  oz.T. 
10.  2.6658+  A.; 
$399.87. 
11.  13fda. 

16.  7815  min. 
17.  4  da.  10  hr.  50  min. 
18.  54738  sec. 

28.  1  pk.  7.4624  qt. 
29.  2  mi.  16  rd.  10  ft. 
6.72  in. 

12.  $1.728. 
13.  3200  mi. 
14.  1810. 

19.  3  wk.  1  da.  21  hr. 

15.  $1.45  per  ream. 

25  mins. 

8.  ¥f  e  yd. 

16.  30  Ib.  12.7  +  oz. 

9.  JYQQ  hr. 

17.  $113.13. 

Page  185. 

10.   2"f"2"o  yr. 

18.  4248  posters  ; 

2.  126000  sec.; 

11       __!  »p 

$27.612,  cost. 

97^00  sec.; 

12.  ^2-  cu.  ft. 

19.  2.6277  bbl. 

76338  sec. 

20.  3  bu.  3  pk.  1  qt. 

3.  123163  sec. 

Page  189. 

1  pt.  too  much. 

4.  130°  9X  20". 

13.  ^f^A. 

21.  67500  Ib. 

5.  106°  48'  20". 
6.  648000;  324000; 

14.  2  Air  bbl- 

Page  193. 

216000. 

6.   Y8j. 

2.  99  Ib.  loz.  15  pwt. 

7.  10800;  7200. 

7.  if. 

3.  £319  Os.  8Jd. 

ANSWERS. 


383 


4.  37  mi.  121  rd.  1  yd. 

3.  159  Ib.    1  oz.   12 

12.  $  .257  +  per.  gal. 

1  ft,  1  in. 

pwt.  15  gr. 

loss. 

5.  25  T.  8  cwt.  40  Ib. 

4.  12  T.  3  cwt.  53  Ib. 

13.  At  $3  per  metre; 

6  oz. 

12  oz. 

$.156  -f-  per  yd. 

6.  47  gal.  1  pt, 
7.  115  bu.  3  pk.  4  qt. 
8.  101  C.    125  cu.  ft, 

5.  127  A.  15  sq.  rd. 
6.  116  rd.  1  yd.  1  ft. 
7.  98  C.  3  cd.  ft.  8 

14.  15.96M. 
15.  1  \  M. 

16.  $46.067  +  . 

518  cu.  in. 

cu.  ft. 

17.  $3500. 

10.  98  yd.  2  ft.  4J-  in. 

8.  £32  19s.  9d. 

18.  89.811  -f  kilos. 

•11.  2  A.  19  sq.  rd.  15 

9.  $49.95. 

19.  18  hectares; 

sq.  yd.    1  sq.  ft, 
18  sq.  in. 

Page  198. 

$4500. 
20.  625  hectolitres. 

12.  109  gal.  1  qt.  1  pt. 

2.  2  bu.  7  qt.  1  pt. 

21.  $29.918. 

3.  85  A.  90  sq.  rd.  6 

22.  11  cents  per  Ib.; 

Page  194. 

sq.yd.  64f  sq.in. 

$11.34. 

13.  15  vr.  7  mo.  18  da. 

4.  78  gal.  3  qt.  1  pt. 

23.  $10.05. 

14.  22  T.  17  cwt.  46  Ib. 

2gi. 

3f  f  oz. 

5.  1  T.  17  cwt. 

Page  208. 

6.  2  C.  118f  cu.  ft. 

1.  TV;  .10. 

2.  316  rd.  1  yd.  1  ft. 

7.  £3  2s.  6fd. 

2.  i;  .125. 

8  in. 

9.  50  times. 

3.  i;  .20. 

10.  4  da.  6  hr.  6  min. 

4.  1;  .25. 

Page  195. 

58  +  sec. 

5.  A;  -30. 

3.  6  cwt.  95  Ib.  2  oz. 

11.  23  mi.  24  rd.  10 

6.  |  ;  .75. 

4.  41  gal.  1  qt.  1  pt. 

ft.  Ill  in. 

7.  |;  .875. 

3gi. 

12.  272%  bbl; 

8.  3-V;  .03125. 

5.  4  Ib.  11  oz.  17  pwt. 

13.  65  spoons;    1  oz. 

9.  TV;  .0625. 

6.  8°  50'  48". 

5  pwt.  left. 

10.  1£;  1.25. 

7.  18  C.  7  cd.  ft.  15 

14.  100  pickets. 

11.  sfo;  .015. 

cu.  ft. 

12.  1;  .331. 

9.  47  sq.  rd.  12  sq.  yd. 

Page  2O1. 

13.  1;  .16f. 

3  sq.ft.  Hlsq.in. 

3.  2  hr.  9  min.  13T95 

14.  21;  2.125. 

10.  £86  16s.  8|d. 

sec. 

15.  A;  .3175. 

11.  148  A.  147  sq.  rd. 

4.  122°  26'  45"  W. 

16.  f  ;  .66-f. 

12  sq.  yd.  5  sq.  ft. 

5.  5hr.  5  min.  32  sec. 

17.  T\;  .1875. 

63  sq.  in. 

6.  5hr.8min. 

18.  A;  .20J. 

12.  6  E.  10  qr.  7  sh. 

7.  11   min.    48   sec. 

19.  ?fo;  -0075. 

14.  8  yr.  2  mo.  12  da. 

past  12  o'clock 

20.  ^;  .005. 

noon. 

21.  ^;  .045. 

Page  196. 

8.  16°  17'. 

22.  T§IO;  .003. 

15.  66  yr  8  mo.  13  da. 

9.  53   min.    50   sec. 

23.  yjfo;  .073. 

16.  10  yr  7  mo.  20  da. 

A.  M.  Jan.  2. 

24.  f  ;  .375. 

17.'  Feb.  13,1841. 

10.  17°  45'  east. 

18. 

Page  21O. 

19.  3yr.  11  mp.  28  da. 

Page  206. 

13.  $7.61. 

20.  5  yr.  3  mo.  1  da. 

8.  40.4671  +  A. 

14.  $6.44. 

9.  ^  hectolitre. 

15.  155  gal. 

Page  197. 

10.  5864.31  dm. 

16.  306  mi. 

2.  69  gal.  3  qt.  1  gi. 

11.  $71.315. 

17.  300  sheep. 

384 


ANSWERS. 


18.  $300. 

29.  213}. 

3.  $39.584. 

19.  $1293.36. 

4.  $5.38. 

20.  $302.64. 

Page  216. 

5.  $3.96. 

21.  $37500. 

9.  400. 

6.  $9.71. 

22.  $2625. 

10.  369. 

7.  $6.19. 

23.  $23200. 

11.  282. 

24.  $22940,  elder  ; 

12.  560. 

Page  223. 

$12333.33J,  y'nger. 

13.  $14000. 

8.  $8.61. 

14.  $300. 

9.  $8.95. 

Page  212. 

15.  $1500. 

10.  $7.917. 

16.  16f%. 

16.  $6000. 

1].  $6.55. 

17.  42i#. 

17.  800. 

18.  31|i%. 

Page  224. 

19.  55f%. 

Page  217. 

2.  $15.22. 

20.  54%. 

10.  525. 

3.  $76.826. 

21.  66f%. 

11.  720. 

4.  $44.91. 

22.  38f%. 

12.  633}. 

5.  $18.835. 

23.  88»%  spent; 

13.  426TV 

6.  $41.54. 

11$%  left. 

14.  $1800. 

7.  $25.12. 

24.  33^%. 

15.  500  bu. 

8.  $47.525. 

25.  $%. 

16.  $41.77}. 

9.  $46.03. 

26.  27T\%. 

17.  620  soldiers. 

10.  $38.61. 

27.  1}%. 

18.  $50000. 

11.  $197.88. 

19.  $13400. 

12.  $92.35. 

Page  213. 

13.  $66.29. 

28.  1|%. 

Page  221. 

14.  $39.807. 

29.  25%;  20%;  18}$. 

14.  $2.26. 

15.  $131.29. 

15.  $5.92. 

16.  $152.40. 

Page  214. 

16.  $8.99. 

17.  $335.938. 

9.  3080. 

17.  $645. 

18.  $339.817. 

10.  2450. 

18.  $735.17. 

11.  8331. 

19.  $773.256. 

Page  226. 

12.  21.45. 

20.  $979.87. 

2.  $49.55. 

13.  $532.50. 

21.  $737.76. 

3.  $408.71. 

14.  343.75  bu. 

22.  $376.309. 

4.  $112.95. 

15.  2100  men. 

23.  $15.96. 

5.  $65.87. 

16.  3300. 

24.  $2.60. 

6.  $139.73. 

17.  366f. 

25.  $6.19. 

7.  $54.61. 

18.  $15000. 

26.  $17.948. 

8.  $84.43. 

19.  $100000. 

27.  $5.55. 

9.  $64.89. 

20.  1102  A.  HOsq.rd. 

28.  $6.28. 

10.  $5.87. 

21.  $1600. 

29.  $8.788. 

11.  $368.96. 

22.  $1882T<y. 
23.  $8000. 

30.  $17.89. 
31.  $624.546. 

Page  228. 

24.  $20000. 

32.  $185.08,  interest. 

2.  $475.44. 

25.  $2700. 

$969.33,  amount. 

3.  $962.32. 

26.  $40206. 

27.  $20000000. 

Page  222. 

Page  229. 

28.  20. 

2.  $64.844. 

1.  $435.55. 

ANSWERS. 


385 


2.  $402.88. 

12.  $297.26. 

19.  $7441.018. 

3.  $91.45. 

Page  237. 

20.  $1300.12  gain. 

Page  231. 

3.  $802.75. 

Page  242. 

2.  $497.80. 

4.  \\%  for  cash. 

2.  $4.137. 

3.  $439.277. 

5.  $2443.74. 

3.  $37.393. 

4.  $4841.52. 

6.  $3290.625. 

4.  $43.748. 

5.  $3986.35. 

7.  $1778.65. 

5.  $22.16. 

6.  $219.31. 

6.  $96.78. 

7.  $15.086. 

Page  239. 

7.  $95.14. 

2.  $894.95,     present 

8.  $49.41. 

Page  232. 

worth  ; 

9.  $8.67. 

8.  $3022.56. 

$80.54  discount. 

10.  $3.179  discount; 

9.  $621.056. 

3.  $777.195,  present 

$378.383  proc'ds. 

10.  $833.818. 

worth  ; 

11.  $10.90  discount  ; 

$68.005  discount. 

$879.34  proceeds. 

Page  233. 

4.  $792.355,  present 

12.  $3.94+. 

4.  2  yr. 

worth  ; 

13.  $138.91  discount; 

•*••  «*  j  *• 
5.  6  mo. 

$166.395  disc't. 

$15587.04  proc'ds. 

6.  2yr. 
7.  4  jr.  8  mo.  14  da. 

5.  $464.717,  present 
worth  ; 

Page  243. 

8.  5  yr.  4  mo.  13  da. 

$111.533  disc't. 

14.  $3729.79. 

9.  16Jyr. 

6.  $7698.31,  present 

15.  $563.80. 

10.  12  J  vr. 
11.  20  yr.;  16f  yr.; 

worth  ; 
$876.68  disc't. 

Page  244. 

14/yr. 

7.  $3948.26,  present 

6.  $1000. 

12.  40  yr.;  33J  yr.; 

28/yr. 

worth  ; 
$325.73  disc't. 

7.  $2000. 
8.  $987.09. 

8.  $2279.79,  present 

9.  $1015.74+. 

Page  234. 

worth  ; 

10.  $1408.11+. 

5.  6%. 

$565.20  disc't. 

11.  $1262.29. 

6.  t%. 

9.  $1586.99,  present 

12.  $5299.46. 

7.  M£. 

wouth  ; 

13.  $1937.56. 

5  /v 

8   7*. 

$165.75  disc't. 

14.  $507.00. 

w«    •  /o« 

9.  7*. 

10.  $4489.07,  present 

15.  $15316.54. 

TA       7/T/ 

worth  ; 

1U.      i%. 

11.  6%. 

$1004.43  disc't. 

Page  245. 

12.  7%. 

11.  $3306.30,  present 

1.  $17500. 

13.  12i^. 

worth  ; 

2.  $140.81   in  favor 

2  /» 

$151.54  disc't. 

of  4%  discount. 

3.  "$305. 

12.  $15547.169. 

3.  $8168.80. 

4.  $178.50. 

13.  $28.44  gain. 

4.  $4137.93. 

5.  $186.18. 

14.  $560.68. 

5.  $4663.39. 

6.  $1473.62. 

6.  $231.77. 

7.  $1399.28. 

Page  24O. 

7.  $6.06+. 

8.  $1571.43. 

15.  $442.52. 

8.  $1969.93. 

9.  $10714.28. 

16.  $296.97. 

10.  $4125. 

17.  $42.85. 

Page  246. 

11.  $376.518. 

18.  $241.94. 

9.  $1186.65. 

25 

386 


ANSWERS. 


10.  $267.07. 

31.  42f  %,  A's  gain  ; 

16.  $3255. 

11.  $8800. 

50T530f%,  B'sgain. 

17.  $3670. 

12.  $9422.22. 

32.  51T6c,%. 

18.  8000  bu.; 

13.  27. 

$66.75  commis'n. 

14.  $9  per  bbl. 

Page  252. 

19.  $1226. 

15.  $24.31. 

34.  $1,  cost  ; 

20.  $1274.625. 

16.  $180.81. 

$1.10,  selPg  price. 

21.  24%. 

17.  $100000. 

35.  $9. 

22.  $2776.19. 

36.  $1. 

Page  248. 

37.  $.40. 

Page  258. 

14.  $50. 

38.  $.55. 

1.  $9. 

15.  $180. 

39.  $100. 

2.  25%. 

40.  $300. 

3.  10%. 

Page  249. 

41.  $2. 

4.  331%. 

2.  $343.75. 

42.  $1.15. 

5.  125%. 

3.  $358.40. 

43.  $37472. 

6.  $3.10. 

4.  $397.81i. 
5.  $51.75. 

Page  253. 

7.  $400. 
8.  $1.50. 

6.  $5.40. 

46.  $.40. 

9.  6%. 

7.  $96.88|. 

47.  $40000. 

10.  IJyr. 

8.  $6480. 

48.  $160869.56. 

11.  50%. 

9.  $3450. 

49.  $.10. 

12.  42?-%. 

10.  $1605. 

50.  $5373. 

13.  Pay  cash;  I32r%. 

11.  $272.50. 

51.  $.50. 

14.  28f%. 

12.  $2.795. 

52.  $2000. 
53.  $4.12g8T. 

Page  259. 

Page  250. 

54.  $3. 

15.  $40. 

13.  $74.75. 

55.  $1.55|. 

16.  6  bbl. 

14.  $422.15|. 

56.  $5.50. 

17.  50%. 

15.  $4.80. 

57.  20%. 

18.  Sell  now  at  20c.; 

16.  .325  ;   .65  ;    .975  ; 

by  $10.49. 

1.1375. 

Page  255. 

19.  $350,  cost  of  the 

18.  33i%. 

2.  $87.56. 

horse  ; 

19.  25$. 

3.  $180. 

$175,  cost  of  the 

20.  28f%. 
21.  20%. 

Page  256. 

carriage. 
20.  $477.01. 

22.  25%. 

5.  $3932.03. 

21.  $15.30    gain     by 

23.  111%. 

6.  14297.3+  yd. 

borrow'g  at  6%. 

24.  12ff%. 

7.  $50.05. 

22.  595.1+  bu. 

8.  $39.21  i 

23.  $200. 

Page  251. 

9.  $19.125. 

24.  $125,  A's  cost; 

25.  2J%. 

$100,  cost  to  nie. 

26.  $2.296  per  box  ; 

Page  257. 

$.875  gain. 

10.  $8.91. 

Page  260. 

27.  1H%. 

11.  $25.088. 

25.  $20407.50. 

28.  66f%. 

12.  $7.084. 

26.  $25777.89. 

29.  12+%. 

13.  $34.828. 

27.  $3288.38. 

30.  For  cash  at  once; 

14.  $1495.09. 

28.  Lost  $6;  4%. 

by  3AV%- 

15.  4557.03+  yd. 

29.  Neither. 

ANSWEES. 


387 


30.  18.96-}-  %. 

5.  $5418.75. 

Page  28O. 

31.  $73.229  gain. 
32.  $120,  selling  price 

6.  $288.75. 
7.  $11407.50. 

2.  $187.50. 
3.  $67.50. 

of  horse  ; 

4.  $270. 

$45,   selling   price 

Page  274. 

5.  $1  359.375,  prem.; 

of  cow. 

9.  25  shares. 

$107390.625,  loss. 

Page  261. 

10.  290  shares. 
11.  600  shares. 

6.  $515.125. 

7.  $55000. 

33.  A,  $200  ; 

12.  34.6  shares. 

8.  2%. 

B,  $1331; 

13.  22.11+  shares. 

C,  $531 

15.  $209.234. 

10.'  $14000. 

34.  40  cents  per  Ib. 

16.  $622.707. 

11.  1-|%. 

35.  $4424.79+ 

17.  5%  at  60  ; 

12.  $9000. 

36.  9^%. 

by  $16.666. 

13.  $2668.75,   loss  of 

37.  $6. 

18.  $641.25. 

the  merchant  ; 

38.  $4.50. 

19.  6%  at  90; 

$22331.25,  insur- 

39. $6267.49; 

by  $1.04. 

ance  co.'s  loss. 

20000  Ib.  wool. 

14.  $13333.331 

40.  $1500. 

Page  275. 

d 

Page  264. 

20.  $109.62  dimin'd. 

Page  281. 

21.  $13500. 

15.  $88400. 

3.  $73.121,  A; 

22.  $15208.33. 

16.  $297000. 

$89.171,  B; 

23.  $26812.50. 

17.  $6666.66f,  stock; 

$63.75,    C. 

24.  $32776.04. 

$13333.33J,  store. 

25.  $12675. 

18.  $15228.42. 

Page  265. 

4.  $12.211  tax; 

Page  276. 

Page  282. 

.003591  +  rate. 

27.  6f%. 

2.  $93.90. 

5.  .002272  +  . 

28.  10%. 

3.  $145.09. 

6.  .007587+  rate; 

29.  6%  stock  at  10% 

4.  $3699.50. 

$83.457  tax. 

discount.                  5.  $1423.45  loss. 

7.  $3826.53,    sum    to 

30.  N.  Y.  7>s;  T8¥°7%. 

6.  $1423.30  loss. 

be  assessed; 

31.  8Jg%. 

7.  $465  loss. 

$1168406+,  value 

33.  214f%. 

of  property. 

34.  25%. 

Page  286. 

35.  70%. 

3.  $1005. 

Page  266. 

36.  85f%. 

4.  $3037.50. 

2.  $178.125. 

37.  225%. 

5.  $4978.75. 

3.  $2.55. 

38.  $9941.20. 

6.  $1471. 

4.  $450. 

39.  $7812.03. 

7.  $4987.50. 

5.  $533.12. 

8.  $3003.75. 

6.  $5248.246. 

Page  277. 

9.  $4928.75. 

7.  $3322.512. 

40.  $4694.84. 
41.  $6888.36. 

Page  287. 

Page  273. 

42.  61%. 

10.  $1486.25. 

2.  $8593.75. 

43.  111%. 

11.  $4952.50. 

3.  $8670. 

44.  Gained  $36.73. 

14.  $5710.72. 

4.  $1595. 

45.  Loss  $26.19. 

15.  $1506.40. 

388                                                 ANSWERS. 

16.  $1213.04. 

Page  300.                  3.  A,  $640; 

Page  288. 

2.  A,  $2100  ; 
B,  §£2100; 

B,  $840; 
C,  $840. 

17.  $10012.51. 

C,  $1800. 

4.  B,  $705; 

18.  $3514.93. 

19.  $1747.81. 

3.  A,  $600  ; 
B,  $1200  ; 
C,  $800. 

C,  $740.25  ; 
D,  $1057.50. 
5.  A,  $426.505  ; 

Page  289. 

4.  A,  $2000  ; 

B,  $621.987; 

2.  £571  4s.  6Jd. 
3.  £732  6s.  2fd. 
4.  £1149  6s.  7£d. 
5.  £1061  7s.  5|d. 
6.  $668.87. 
7.  $1832.25. 

'  B,'$1600; 
C,  $2400. 
5.  A,  $200; 
B,  $160; 
C,  $280. 
6.  A,  $600; 
B  $750- 

C,  $426.505. 
6.  A,  $2550  ; 
B,  $3400; 
C,  $2550. 

7.  A,  $1080; 
B,  $1600; 
C,  $1820. 

Page  29O. 

8.  7621.67  fr. 
9.  $1169.77. 
10.  $4650. 
11.  $3063.14. 
12.  $4484.11. 

C,$675; 

D,  $975. 
7.  D,  $1600; 
G,  $2000; 
L,  $1800. 
8.  E,  $862.50  ; 
.       F,  $575; 

8.  G,  $.561.  702; 
L,  $702.127; 
F,  $936.170. 

Page  31O. 

3.  $125. 

4.  $8.75. 

Page  292. 

G,  $862.50. 

5.  $45. 

9.  A,  $2744.78  ; 

6.  26£  T. 

2.  2  mo.  29  da. 

B,  $2299.33  ; 

7.  24T6T. 

3.  2  mo.  18  da. 

C,  $1446.63. 

8.  4TV 

4.  1  mo.  12  da. 

10.  A,  $74.86; 

9.  12  men. 

5.  2  mo.  10  da. 

B,  $86.84; 

10.  9T\  da. 

6.  3J  mo. 

C,  $104.81  ; 

11.  36ff  bu. 

Page  294. 

D,  $203.63. 

12.  1000  bbl. 
13.  40  J  da- 

2.  June  20,  1877. 

Page  301. 

3.  May  2,  1877. 
4.  Jan.  24,  1877. 
5.  Dec.  22,  1876. 
6.  April  23,  1877. 
7.  June  7,  1877. 
8.  Aug.  16,  1877. 

11.  A,  $1200  gain; 
B,  $1600  gain; 
C,  $7000  stock. 
12.  A,  $335.365; 
B,  $402.439  ; 
C,  $536.585  ; 

Page  311. 

14.  520  bu. 
15.  2£J  A. 
16.  $1628.25. 
17.  65  da. 

18.  9792f  Ib. 

Page  296. 

D,  $670.731  ; 
E,  $804.878. 

19.  2307.V  mi. 
20.  162TV.Tmi. 

2.  Aug.  22,  1877. 

13.  A,  $750; 

21.  427|rd. 

3.  Mar.  21,  1877. 

B,  $1000; 

22.  4fJM 

C,  $1250. 

23.  20  da. 

Page  297. 

4.  June  19,  1877. 

Page  3O2. 

Page  313. 

5.  June  15,  1877. 

2.  A,  $2880  ; 

2.  28J  da. 

6.  July  5,  1877. 

B,  $3600  ; 

3.  17820  Ib. 

7.  $1198.60. 

C,  $2880. 

4.  $6875. 

ANSWERS. 


389 


5.  $710.76  +  . 

9.  821. 

Page  333. 

6.  19TyT  da. 

10.  886. 

3.  42. 

11.  969. 

4.  64. 

Page  314. 

12.  2424. 

5!  55*. 

7.  $594. 

13.  3546. 

6.  89. 

8.  $902.77. 

14.  5555. 

7.  57. 

9.  473  vd. 

15.  472;  3375. 

8.  63. 

10.  8111  bu. 

16.  .874;  .5555. 

9.  177. 

11.  25|f. 

10.  126. 

12.  21  da. 

Page  325. 

11.  536. 

13.  27600  Ib. 

17.  .306. 

12.  1.259+. 

14.  18750  Ib. 

18.  .315. 

13.  2.0800+. 

15.  21J-T. 

19.    23^ 

14.  .6463+;  .8617+. 

16.  15f  da. 

20.  fff. 

15.  .8739+;    .849+; 

17.  546i23  ft. 

O1       942 

AV 

Page  316. 

22*.  707*10  +  . 
23.  .86602  +  . 

1.  45  ft. 

2.  1728;  12167; 

24.  .79056  +  . 

2.  24  in. 

59319;  13824. 
3.  2209  ;  2601  ; 

25.  .9486  +  . 

Page  334. 

841;  1156. 

1.  25  ft. 

3.  13  ft. 

4.  225;  1089;  576; 

2.  45  rd. 

4.  10.75  ft. 

1296;  625. 

3.  52  ft.  wide  ; 

5.  10.82  ft. 

5.  21952;  91125; 

104  ft.  long. 

6.  21.50  ft. 

5832;  9261;  68921. 

4.  480  rd. 

7.  12.9  +  in. 

'•    12Y  )    T2~l>  )    7~2  i)  J 

5.  240  rd. 

8.  19.37  in. 

6.  $44.      • 

9.  $19.51. 

8.  i^Vr*'  A2A- 

10.  3.46+  ft,  width; 

9!  50625. 

Page  327. 

10.38+  ft,  leng  h 

10.  15625. 

2.  25  ft. 

11.  Rectangle  ; 

11.  27000. 

3.  113.137  ft. 

143.55  +  sq.  ft. 

12.  .00000625. 

4.  40  ft. 

12.  39.37  in. 

13.  .000000125. 

5.  122.474  ft. 

14.  4.2025. 

6.  140.584  ini. 

Page  335. 

15.  J44. 

2.  10ft. 

•     2  8  9* 

Page  328. 

3.  $3375. 

17!  9lf. 

7.  75  rd. 

4.  10.75  ft. 

18.  641J. 

8.  119.482ft. 

5.  2.38. 

19.  9.0200V 

9.  205.704  ft. 

6.  Twice  as  great. 

20.  20.251285||. 

10.  172.046  ft. 

7.  27  times. 

21.  10000;  512;  729. 

11.  386.003  ft. 

8.  1st,  .506; 

2d,  .721; 

Page  324. 

Page  329. 

3d.  2.773. 

3.  53. 

2.  25.1328. 

9.  23.48. 

4.  63. 

4.  63.245. 

5.  66. 

5.  12  ft. 

Page  338. 

6.  96. 

6.  100  rd.  length; 

2.  55. 

7.  266. 

10  rd.  breadth. 

3.  198. 

8.  344. 

7.  15.94268  rd. 

4.  $1.72. 

390 


ANSWERS. 


5.  209TV  ft. 

Page  347. 

6.  $4013.837. 

6.  5070. 

1.  19.635  sq.  ft. 

8.  3775. 

2.  50.265  sq.  ft. 

Page  354. 

9.  505. 

3.  1145.9  sq.  rd. 

1.  84.8232  cu.  ft. 

10.  330  mi. 

4.  795.77  sq.  ft. 

2.  18000  cu.  ft. 

11.  78. 

5.  50.929  A. 

3.  12440.736  Ib. 

12.  $3360. 

6.  706.86  sq.  rd. 

4.  7296  Ib. 

2.  2430. 

7.  7.136  rd. 
8.  12  rd. 

1.  4666f  cu.  ft. 

Page  34O. 

9.  2  acres  33.3939 

Page  355. 

3.  10240. 
4.  $984.15. 

5.  $133.82+. 
6.  $696.849. 
8.  1364. 

sq.  rd. 
10.  962.115  sq.  ft, 

Page  350. 

1.  31.416  sq.  ft. 

2.  236.405+  en.  ft 
3.  136.136+  cu.  ft. 
4.  6415.28  gal. 

1.  65.45  cu.  ft. 

9.  4JfJ. 
10.  4. 

Page  343. 

2.  40  sq.  ft. 
3.  144  sq.  ft. 
4.  37.6992  sq.  ft. 
5.  63  sq.  ft. 

2.  268.0832  cu.  ft. 
3.  14.1372  cu.  ft. 
4.  795.217+  Ib. 
5.  8.181  cu.  ft. 

1.  31.416  ft. 
2.  141.372  ft. 

Page  351. 

6.  8181.25  cu.  ft. 

3.  2  mi.  302.48  rd. 

1.  540  sq.  ft. 

Page  356. 

4.  125.664  rd. 

2.  376.992  sq.  ft. 

1.  10  da. 

3.  628.32  sq.  ft. 

2.  $12. 

Page  344. 

4.  $64. 

3.  80  men. 

5.  34.557ft. 
6.  101.38  rd. 
7.  204.354  rd. 

5.  89.5356  sq.  ft. 
6.  400  sq.  ft. 
7.  75.3984  sq.  ft. 
8.  157.08  sq.  ft. 

4.  5|  mi. 
5.  16  men. 

6.  $5k 

7.  $75. 

1.  520  sq.  ft. 
2.  25  1  sq.  ft. 

Page  352. 

8.  336  Ib. 
9.  $125.635. 

3.  720  sq.  rd. 

1.  251.328  sq.  ft. 

10.  13Joz. 

4.  525  sq.  ft. 

2.  3000  sq.  ft. 

3.  $5.654. 

11.  60  sheep. 

Page  345. 

4.  340  sq.  ft. 

Page  357. 

1.  216  sq.  ft. 

5.  $26.88. 

12.  $80. 

2.  126  sq.  ft. 
3.  $168.376. 
4.  5935.85  sq.  ft. 

5.  $8.64. 
6.  15000  sq.  ft. 

1.  4.908  sq.  ft. 
2.  1.396  sq.  ft. 
3.  26.50+  sq.  in. 

Page  353, 

13.  5J}. 
14.  $40.33  loss. 
15.  451  trees. 
16.  $48,  A's  money; 
$40,  B's  money. 

Page  346. 

4.  45.836  sq.  ft. 

17.  $65,  A's; 
$50,  B's  ; 

1.  5500  sq.  ft. 

1.  2  on.  ft. 

$55,  C's. 

2.  54  sq.  rd. 

2.  7.0686  cu.  ft. 

18.  48^  ft. 

3.  576  sq.  ft. 

3.  $9. 

19.  A's,  105  Ib.; 

4.  $546.875. 

4.  462.857+  bu. 

B's,  1401b.; 

5.  $2062.50. 

5.  2632.089+  gal. 

C's,  245  Ib. 

ANSWERS. 


391 


20.  A's,  $32; 

Page  35£>. 

58.  B's  age,  60  yr.; 

B's,  $27. 

33.  72  mi. 

C's  age,  80  yr. 

21.  34,    A's  age  ; 

34.  200. 

59.  $3000. 

45^,  B's  age  ; 

35.  57T3T. 

60.  $80. 

56f  ,  C's  age. 

36.  15. 

61.  2  cts.,  apples; 

37.  30  da. 

3  cts.,  pears. 

Page  358. 

38.  $2£. 

82.  A,  46|  da.; 

22.  720  apples. 
23.  18f  §f  da.,  time  in 

39.  15TV  yd. 
40.  50|  oz. 

B,  35  da. 

63.  $1186.98. 

which  all  can  do 

41.  566  tiles. 

64.  $40  loss; 

it; 

42.  $2875. 

6J%. 

58|f  da.,  time  in 

43.  26}. 

65.  27.64  ft. 

which  A  can  do 

44.  27T3y  min.  past  5. 

66.  15  yr. 

it; 
70Jf  da.,  time  in 

45.  6  mi.  per  day. 
46.  30  men. 

Page  362. 

which  B  can  do 

67.  90  Ib. 

it; 

Page  36O. 

68.  162  in  No.l; 

46T7245  da.,  time  in 

47.  20  min.  past  5. 

144  in  No.  2; 

which  C  can  do 

48.  $882.46. 

128  in  No.  3. 

it. 

49.  $74.07. 

69.  A,  in  24  da.; 

24.  6f  da. 

50.  $740.52+. 

B,  in  17f  da.; 

25.  $240,  one; 

51.  16f  %  gain. 

C,  in  40  da. 

$150,  other. 

52.  42f%. 

70.  A's  share,  $3; 

26.  810  revolutions. 

53.  $270,  carriage  ; 

B's  share,  $21. 

27.  72  ft. 

$240,  horse. 

71.  92160  A. 

28.  $3.80. 

54.  300  cats. 

72.  21. 

29.  8  hr.  48  min. 

55.  $24. 

73.  $600. 

30.  $208.33i. 

56.  $500,  cost. 

74.  $4563,  son's; 

31.  $1.50,  wheat; 

$1521,  widow's; 

$  .40,  corn. 

Page  361, 

$507,  daughter's. 

32.  900  rd. 

57.  7  da. 

UNIVEESITY  OF  CALIFORNIA  LIBRARY, 
BERKELEY 


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